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1. ## series convergence
Hi.
series: Sigma n from 1 to infinity arctan(n) / n^3/2
I want to determine convergence/divergence.
What i tried:
f(x) = arctan(x) / x^3/2
arctan(x) > f(x)
since arctan(x) converges to pi/2 as x -> inf, f(x) should converge too.
Is that reasoning right?
2. Originally Posted by Kuma
series: Sigma n from 1 to infinity arctan(n) / n^3/2 I want to determine convergence/divergence.
$\dfrac{{\arctan (n)}}
{{\sqrt[3]{{n^2 }}}} \leqslant \dfrac{{\frac{\pi }
{2}}}
{{\sqrt[3]{{n^2 }}}}$
|
|
# On Sample Complexity of Projection-Free Primal-Dual Methods for Learning Mixture Policies in Markov Decision Processes
We study the problem of learning policy of an infinite-horizon, discounted cost, Markov decision process (MDP) with a large number of states. We compute the actions of a policy that is nearly as good as a policy chosen by a suitable oracle from a given mixture policy class characterized by the convex hull of a set of known base policies. To learn the coefficients of the mixture model, we recast the problem as an approximate linear programming (ALP) formulation for MDPs, where the feature vectors correspond to the occupation measures of the base policies defined on the state-action space. We then propose a projection-free stochastic primal-dual method with the Bregman divergence to solve the characterized ALP. Furthermore, we analyze the probably approximately correct (PAC) sample complexity of the proposed stochastic algorithm, namely the number of queries required to achieve near optimal objective value. We also propose a modification of our proposed algorithm with the polytope constraint sampling for the smoothed ALP, where the restriction to lower bounding approximations are relaxed. In addition, we apply the proposed algorithms to a queuing problem, and compare their performance with a penalty function algorithm. The numerical results illustrates that the primal-dual achieves better efficiency and low variance across different trials compared to the penalty function method.
• 5 publications
• 5 publications
• 4 publications
• 31 publications
10/17/2017
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## 1. Introduction
This manuscript is the (extended) conference paper.
M
arkov decision processes are mathematical models for sequential decision making when outcomes are uncertain. The Markov decision process model consists of decision epochs, states, actions, transition probabilities, and costs. Choosing an action in a state generates a cost and determines the state at the next decision epoch through a transition probability function. Policies or strategies are prescriptions of which action to take in a given state to minimize the cost. Given a MDP, the main objective is to compute (near-) optimal policies that (approximately) attain the minimum long-term average cost.
In most practical applications, the underlying MDP is compactly represented and the number of states scale exponentially with the size of the representation of the MDP. In addition, for such applications, various hardness results often indicate that computing actions of optimal policies is intractable in the sense that polynomial-time algorithms to compute such control policies are unlikely (unless come complexity class collapse) or simply don’t have guarantees for searching policies within a constant multiplicative or additive factor of the optimal; see, e.g., [1]. In view of those negative results, it is natural to pursue a more modest objective which is to compute the actions of a policy that is nearly as good as a policy chosen by an oracle from a given restricted policy class. Following the work of [1], in this paper we consider the policy class to be the convex-hull of a set of known base policies. Such base policies are often known in practice for certain applications. For instance, a myopic and a look ahead policy in queuing networks can be combined to achieve a mixture policy with better performance.
### 1.1. Main Contributions
The main contributions of this paper are summarized as follows:
• We formulate the optimization over the restricted class of mixture policies as an approximate linear programming (ALP) for MDPs, where the feature vectors are given by the occupation measures of the base policies.
• We propose a novel projection-free
stochastic primal-dual (SPD) algorithm for the reinforcement learning of efficient mixture policies in Markov decision processes.
• We analyze the constraint violation of the solutions of SPD, and prove that such constraint violation diminishes in the asymptotic of many rounds.
• We analyze the sample complexity of the proposed algorithm, i.e., the number of queries required from a sampling oracle to achieve near optimal cost function.
• We numerically compare the performance of the proposed primal-dual algorithm with that of the penalty function method for a queuing problem, and we show that the solutions obtained by the proposed method in this paper has a smaller variance across different trials compared to the penalty function method.
### 1.2. Connections to Prior Works
The ALP as a framework to find a “low-dimensional” representation of “high-dimensional” functions on a state (action) space has a long history in decision theory; see, e.g., [2, 3, 4, 5, 6] and references therein. The seminal work of De Farias and Von Roy [3]
studies an ALP for stochastic control problems as a means of alleviating the curse of dimensionality. The main challenge in using the proposed ALP is that while it has a relatively small number of variables, it has an intractable number of constraints. To address this issue, the same authors proposed a constraint sampling scheme in a separate work
[7]. In the same line of work, a dual approximate linear programming for the MDP is considered in [5], where the optimization variable is a stationary distribution over state-action pairs. A neighborhood of a low-dimensional subset of the set of stationary distributions defined in terms of state-action features is considered as the comparison class. In a similar line of work, a -learning algorithm is proposed in [4] which leverages state and action features to reduce the dimensionality of the state and action spaces. In addition, the sample complexity of the proposed -learning algorithm is analyzed.
Our work is also closely related to the recent study of Banijamali, et al. [1]. Therein, the authors propose a stochastic gradient decent in conjunction with a penalty function method to optimize over a set of mixtures of policies. The main challenge in using a penalty function method is that its performance is often sensitive to the choice of the penalty factor in addition to the learning rate. Moreover, it yields a large constraint violation as is observed in [8, Thm. 1]. Furthermore, to optimize the regret performance , the authors in [1] propose a penalty factor that depends on the amount of violation of constraints, which is unknown in practice. In this paper, we propose a stochastic primal-dual method whose only hyper-parameter is the learning rate.
We also mention the recent work of Chen and Wang [4], where a primal-dual algorithm with the Bregman divergence is considered in conjunction with the approximate linear programming formulation of MDPs. The work of [4] deals with the sample complexity of the stochastic primal-dual algorithm, namely the number of queries to the oracle needed to attain near optimal value function, whereas in this paper we are concerned with the efficiency in the dual space as well as the constraint violation of the primal-dual solutions. In addition, the algorithm we propose differs from [4] in several key points. First, unlike the algorithm of Chen and Wang, our approach does not require the realizability assumption (cf. [4, Def. 1]). The realizability condition requires the spanning vectors of features in ALP to be non-negative, which in turn allows one to simplifies the projection onto the simplex section of a hyper-plane. In particular, projection onto the simplex can be implemented efficiently using the Kullback-Leibler (KL) divergence as the proximal function. In our algorithm, we provide a direct method of projection onto the hyper-plane which obviates the realizability condition and provides a more expressive representation. In addition, we present a randomized policy based on the primal-dual solutions that differs from the policy proposed in [4]. Second, our proposed algorithm solves an optimization problem in the dual space, where feature vectors are the occupation measures of a set of base policies. This allows us to compute a randomized policy directly from the solution of the underlying dual optimization problem. Lastly, the role of the Bregman divergence in our algorithm is to implicitly enforce the constraints due to the size of the policy class. In contrast, the Bregman divergence in [4] is used as a mean to attain better sample complexities via adaptation to the geometry of the feasible set.
### 1.3. Paper Outline
The rest of this paper is organized as follows. In Section 2, we present preliminaries related to MDPs. In addition, we review the approximate linear programming formulation of MDPs based on the linear representation of large state-spaces. In Section 3, we present a stochastic regularized primal-dual algorithm to compute the coefficients of the mixture policy. We also present the main theorem concerning the performance of the proposed algorithm. In Section 5 we present the proof of the main theorem, while deferring technical lemmas to the appendices. Lastly, in Section 7, we conclude this paper.
## 2. Preliminaries
In this section, we first present the notations and technical definitions that we need to establish our theoretical results. We then review relevant definitions regarding infinite horizon discounted Markov decision processes. We also review Bellman’s equations as well as linear programming dual formulation of Bellman’s equations.
Notations and Definitions. We denote vectors by the lower case letters (e.g.
), and random variables and matrices with the upper case letters (
e.g. ). The dual norm conjugate to the norm is defined by . We denote the Euclidean projection by . We use the standard asymptotic notation with the following definition: If are two functions from to , then if there exists a constant such that for every sufficiently large and that if . For positive integer , we use the shorthand to denote the set . For a matrix , let denote the subordinate norm for
. We denote the largest singular value by
. Further, we use the shorthand notations , , and . A function is -Lipschitz with respect to the norm over iff
|f(x)−f(y)|≤L∥x−y∥,for all x,y∈X.
A function is -smooth with respect to the norm over iff
∥∇f(x)−∇f(y)∥∗≤β∥x−y∥,for all x,y∈X.
A function is -strongly convex with respect to the norm over iff
f(x)+⟨g,y−x⟩+μ2∥x−y∥2≥f(y),
for all . The effective domain of a function is the following set . The sub-differential set of a function at the point is defined as follows
(2.1) ∂f(x0)def={g:f(x)−f(x0)≥⟨g,x−x0⟩,∀x∈dom(f)}.
The relative interior of a convex set , abbreviated , is defined as , where denotes the affine hull of the set , and is a ball of radius centered on .
The Fenchel conjugate of the function is defined as follows
(2.2) f∗(y)=supx∈X{⟨x,y⟩−f(x)}.
###### Definition 2.1.
(Orlicz Norm) The Young-Orlicz modulus is a convex non-decreasing function such that and when . Accordingly, the Orlicz norm of an integrable random variable with respect to the modulus is defined as
(2.3) ∥X∥ψdef=inf{β>0:IE[ψ(|X|−IE[|X|]/β)]≤1}.
In the sequel, we consider the Orlicz modulus . Accordingly, the cases of and norms are called the sub-Gaussian and the sub-exponential norms and have the following alternative definitions:
###### Definition 2.2.
(Sub-Gaussian Norm) The sub-Gaussian norm of a random variable , denoted by , is defined as
(2.4) ∥Z∥ψ2=supq≥1q−1/2(IE|Z|q)1/q.
For a random vector , its sub-Gaussian norm is defined as
(2.5) ∥Z∥ψ2=supx∈Sn−1∥⟨x,Z⟩∥ψ2.
###### Definition 2.3.
(Sub-exponential Norm) The sub-exponential norm of a random variable , denoted by , is defined as follows
(2.6) ∥Z∥ψ1=supq≥1q−1(IE[|Z|q])1/q.
For a random vector , its sub-exponential norm is defined as
(2.7) ∥Z∥ψ1=supx∈Sn−1∥⟨Z,x⟩∥ψ1.
###### Definition 2.4.
(Legendre Function) A function , is called a Legendre function (a.k.a. essentially smooth functions) if it satisfies the following conditions:
• and and is convex.
• is strictly convex.
• partial derivatives of exists and are continuous for all .
• Any sequence converging to a boundary point of satisfies
limn→∞∥∇ϕ(xn)∥=∞.
###### Definition 2.5.
(Bregman Divergence) Suppose , is a Legendre function. The Bregman divergence is defined as follows
Dϕ(x;y)def=ϕ(x)−ϕ(y)−⟨x−y,∇ϕ(y)⟩,
where is the gradient vector evaluated at .
### 2.1. Markov Decision Processes (MDPs)
In this paper, we consider MDPs with high dimensional state and action spaces. The first definition formalizes MDPs:
###### Definition 2.6.
(Markov Decision Process) A Markov decision process (MDP) is a 6-tuple consists of:
• Decision epochs: The set represents the set of times at which decisions are to be made. If is finite, then the MDP is said to be a finite horizon MDP with -epochs. If , the MDP is said to be an infinite horizon MDP.
• States: We assume that the state-space is finite.
• Actions: We assume that the action set is also finite.
• Transition model: The transition model for each and
, is the probability distribution
on . The element represents the probability of transitioning to the state after performing the action in the state . We define the matrices and .
• Cost function: For each , is the cost function. Let , and .
• Discount factor: The discount factor reflects the intertemporal preferences.
We consider discounted infinite horizon MDP characterized by the tuple . For such MDPs, we compute randomized policies which we formally define below:
###### Definition 2.7.
(Randomized Policy) A randomized policy is the sequence of distributions, where is a probability distribution on the action space , conditioned on the state . The value represents the probability of taking the action in the state . Let denotes the set of all such randomized policies.
The objective of discounted infinite horizon MDP is to find a policy such that the infinite-horizon sum of discounted costs is minimized regardless of the initial state , i.e.,
minπ∈Πvπ(s)=liminfT→∞IEπ[T∑t=0γtcat(st)∣∣∣s0=s],
where and are the realizations of state transitions and actions, respectively, generated by the Markov decision process under a given policy , and the expectation is taken over the entire trajectory.
Define the Bellman operator for all . From the theory of dynamic programming, a vector is the optimal value function to the MDP if and only if it satisfies the following Bellman fixed point equation [9]
(2.8) Sv∗=v∗,
where is the difference-of-value vector. A stationary policy is an optimal policy of the MDP if it attains the element-wise maximization in the Bellman equation (2.8), i.e., .
Alternatively, the Bellman equation (2.8) can be recast as the following linear programming problem (cf. [10]),
(2.9a) maxv∈IRn+ αTv (2.9b) s.t.:(In−γPa)v−ca⪰0,∀a∈A,
where , and is the initial distribution over the states, and is the simplex of the probability measures on the set of states .
To characterize the dual problem associated with the primal problem (2.9), Let denotes the stationary distribution under the policy and initial distribution of states . In particular, let denotes the transition probability matrix induced by a fixed policy whose matrix elements are defined as . Alternatively, let be a matrix that encodes the policy , i.e., let , where other entries of the matrix are zero. Then, .
Furthermore, let
denotes the stationary distribution of the Markov chain induced by the policy
, i.e., , where
(2.10) μπα =(1−γ)α∞∑t=0γt(Pπ)t (2.11) =(1−γ)α(In−γPπ)−1.
The measure captures the expected frequency of visits to each state when the underlying policy is , conditioned on the initial state being distributed according to . Future visits are discounted by the factor .
We define the occupation measure as the vector defined by , where is the Hadamard (element-wise) vector multiplication.. Then,
π∗ =argminπ∈Π∑s∈Sμπα(s)∑a∈Aπa(s)ca(s) =argminπ∈Π∑s∈S∑a∈Aξπa(s)ca(s) (2.12) =argminπ∈ΠcTξπ.
Thus, the dual problem associated with the primal problem in Eq. (2.9a) has the following form
(2.13a) minξ∈IRmncTξ (2.13b) s.t.:ξT(P−Q)=0, (2.13c) 0⪯ξ,ξT1=1.
where is a binary matrix such that the -th column has ones in rows to , and . In the optimization problem in Eq. (2.13), the constraint (2.13c) ensures that is a distribution, and the constraint (2.13b) guarantees that is stationary.
Let denotes the optimal solution of the linear programming problem in Eq. (2.13). An optimal randomized optimal policy can be characterized as follows
(2.14) π∗a(s)=ξ∗a(s)μπ∗α(s)=ξ∗a(s)∑a∈Aξ∗a(s),∀a∈A,∀s∈S.
Furthermore, the optimal objective value of the linear programming problem in Eq. (2.13) corresponds to the optimal discounted-cost value under the optimal policy and the initial distribution of the states.
### 2.2. Approximate Linear Programming (ALP) for the Linear Representation of Large State Spaces
It is well-known that the sample complexity as well as computational complexity of solving the linear programming dual problem in Eq. (2.13) scales (at least) linearly with the size of the state space , rendering exact representation intractable in the face of problems of practical scale; see, e.g., [11]. Consequently, to deal with MDPs with a large state space, it is practical to map the state space to a low dimensional space, using the linear span of a small number of features.
Specifically, let denotes a matrix whose column vectors represent feature vectors. In the feature space, the distribution is spanned by the linear combination of features , namely, . Here, the radius and the general norm determine the size and geometry of the policy class , respectively. The dual optimization problem in Eq. (2.13) can thus be reformulated in terms of features
(2.15a) minθ∈ΘRcTΨθ (2.15b) s.t.:θTΨT(P−Q)=0, (2.15c) 0⪯Ψθ,θTΨT1=1.
Designing the feature matrix for Markov decision processes is a challenging problem. In particular, we wish to design a feature matrix such that the linear expansion is expressive, but does not lead to over-fitting. In this paper, we focus on the set of features associated with a set of known base policies. Formally, we consider the set of -base policies and define the subset as the convex hull of base policies,
(2.16) Πddef={πω:πω=d∑i=1ωiπi,d∑i=1ωi=1,ωi≥0,i=1,2,⋯,d}.
Corresponding to each base policy , a stationary distribution is defined according to Eq. (2.11). The dual space of in Eq. (2.16) is then defined as the linear combinations of occupation measures
(2.17) Ξddef={ξθ:πω=d∑i=1ωiπi,d∑i=1ωi=1,i=1,2,⋯,d}.
For all the state-action distribution is defined as for all . With this choice of the feature vectors, we have and as the columns of the matrix are stationary probability distributions. Therefore, the dual optimization (2.15) takes the following form
(2.18a) minθ∈ΘRcTΨθ (2.18b) s.t.:0⪯Ψθ,θT1=1.
Let and . The feasible set of the dual optimization in Eq. (2.18) is the intersection of the following sets
(2.19) Θd=Θd1∩Θd2∩ΘdR.
Let denotes an approximate solution of the optimization problem in Eq. (2.15) generated by an (arbitrary) optimization algorithm after rounds. When is a feasible solution of Eq. (2.15), then defines a valid occupation measure. However, in this paper we permit the feature vectors to violate the non-negativity constraint defined by in the following sense: let denotes a function that quantifies the amount by which the vector violates the non-negativity constraints. In particular, iff . After rounds of the algorithm we propose in the next section (cf. Algorithm 1
), it outputs an estimate
that may potentially violate the constraints defined by , and thus . Nevertheless, we impose the constraint that the estimate generated by the algorithm satisfies the constraints in the asymptotic of many rounds .
By allowing such constraint violation, we devise an efficient algorithm whose only projection is onto the hyper-plane section of the feasible set , and the constraint due to the size of the policy class is enforced implicitly using a Bregman divergence whose domain is subsumed by the set . As we discuss in the next section, the Bregman projection onto the hyper-plane has an efficient implementation.
Notice, however, that due to the (potential) constraint violation of solutions , the vector may no longer be a valid occupancy measure. Nonetheless, it defines an admissible randomized policy via the following relation
(2.20) πˆθTa(s)=[(ΨˆθT)a(s)]+∑a∈A[(ΨˆθT)a(s)]+,∀a∈A,∀s∈S.
In the case that for all pairs of action-state , we define
to be the uniform distribution. Let
denotes the occupancy measure induced by the policy defined in Eq. (2.20), i.e., .
### 2.3. Expanded Efficiency
Equipped with the dual optimization problem in Eq. (2.15), we now describe the notion of efficiency of an algorithm to solve (2.15). The following definition is adapted from [5]:
###### Definition 2.8.
(Efficient Large Scale Dual ALP [5]) For an MDP specified by the cost matrix , probability transition matrix , a feature matrix , the efficient large-scale dual ALP problem is to produce an estimate such that
(2.21) cTξˆθ≤minθ∈ΘdcTξθ+O(ε),
in time polynomial in and under the model of computation in (A.3).
As described by Definition 2.8, the computational complexity depends on the number of features only, and not the size of state-space .
The preceding definition is a special case of the following generalized definition that allows for the constraint violation:
###### Definition 2.9.
(Expanded Efficient Large Scale Dual ALP [5]) Let be some violation function that measures how far is from a valid stationary distribution. In particular, iff is a feasible point for the dual ALP in Eq. (2.13). Then,
(2.22) cTξˆθ≤minθ∈Θd2∩ΘdR[cTξθ+1εV(θ)]+O(ε),
in time polynomial in and , under the model of computation in (A.3).
Clearly, a guarantee for (2.22) implies a guarantee for Eq. (2.21). In addition, the expanded problem has a larger feasible set. Therefore, even if many feature vectors may not admit any feasible points in the feasible set and the dual problem is trivial, solving the expanded problem is still meaningful.
### 2.4. Assumptions
To establish our theoretical results, we need the following fast mixing assumption on underlying MDPs. This assumption implies that any policy quickly converges to its stationary distribution:
• (Fast Mixing Markov Chain) For , the Markov decision process specified by the tuple is -mixing in the sense that
(2.23) τmix(ε)def=maxπ∈Πmin{t≥1:∥(Pπ)t(s,⋅)−μπα∥TV≤ε,∀s∈S},
for all , where for two given Borel probability measures , is the total variation norm.
The fast mixing condition (2.23) is a standard assumption for Markov decision processes; see, e.g., [5], [12]. The fast mixing condition (2.23) implies that for any policy , there exists a constant such that for all the distributions over the state-action space,
(2.24) ∥νPπ−ˆνPπ∥TV≤e−1τmix(ε)∥ν−ˆν∥TV.
The following assumption is also standard in the literature; see, e.g., [12]:
• (Uniformly Bounded Ergodicity) The Markov decision process is ergodic under any stationary policy , and there exists such that
(2.25) 1n√κ1≤μπα≤√κn1.
where we recall from Section 3.1 that is the stationary distribution over the state space of the MDP under the policy , and with the initial distribution on the states.
Under the condition (2.25) of (A.2), it is well-known that the underlying MDP is unichain [13], i.e., a Markov chain that contains a single recurrent class and possibly, some transient states. Moreover, the factor determines the amount of variation of stationary distributions associated with different policies, and thus can be sought of as a form of measuring the complexity of a MDP. Notice that in the case that some policies induce transient states (so the stationary distribution is not bounded away from zero), their mixture with an ergodic policy guarantee ergodicity.
Lastly, we consider the setup of the reinforcement learning in which the cost function is unknown. But, the agent can interact with its environment and receives feedbacks from a sampling oracle:
• (Model-Free Reinforcement Learning) We consider the following model of computation:
1. [leftmargin=*]
2. The state space , the action spaces , the reward upper bound and lower bounds and , and the discount factor are known.
3. The cost function is unknown.
4. There is a sampling oracle that takes input and generates a new state with probabilities and returns the cost .
## 3. Stochastic Primal-Dual Proximal Algorithm
In this section, we first describe a stochastic primal-dual proximal method to compute the coefficients of the mixture model in the dual optimization problem in Eq. (2.18). We then analyze the efficiency as well as the sample complexity of the proposed algorithm.
### 3.1. Stochastic Primal-Dual Algorithm
To apply the stochastic primal-dual method, we recast the optimization problem (2.18) as the following MinMax optimization problem
(3.1) minθ∈Θd2∩ΘdRmaxλ∈IRnm+LΨ(θ,λ)def=cTΨθ−λTΨθ.
The following definition characterizes the notion of the saddle point of a MinMax problem:
###### Definition 3.1.
(Saddle Point of MinMax Optimization Problem) Let denotes a saddle point of the MinMax problem in Eq. (3.1), i.e., a point that satisfies the inequalities
(3.2) LΨ(θ∗,λ)≤LΨ(θ∗,λ∗)≤LΨ(θ,λ∗),
for all and .
The primal optimal point of the MinMax problem (3.1) is an optimal solution for the dual problem (2.18).
Applying the stochastic primal-dual method to the MinMax problem (3.1) is challenging as the the Lagrange multipliers may take arbitrarily large values during the iterations of the primal-dual algorithm, resulting in a large sub-gradients for the Lagrangian function and instability in the performance. The following lemma, due to Chen and Wang [14, Lemma 1], provides an upper bound on the norm of the optimal Lagrange multipliers:
###### Lemma 3.2.
(Upper Bound on the Lagrange Multipliers, [14, Lemma 1]) Suppose is a Lagrange multiplier vector satisfying the MinMax condition in Eq. (3.2). Then, , and for all .
Now, define the norm ball , where . Here, is a free parameter of the algorithm that will be determined later.
Lemma 3.2 suggests that the vector of optimal Lagrange multipliers belongs to the compact set . Therefore, instead of the MinMax problem (3.1), we can consider the following equivalent problem
(3.3) minθ∈Θ2∩ΘRmaxλ∈ΛLΨ(θ,λ),
where the Lagrange multipliers are maximized over the compact set instead of entire non-negative orthant . As we alluded earlier, the set of saddle points of the MinMax problems in Eqs. (3.3) and (3.1) coincide.
Algorithm 1 describes a procedure to solve the MinMax problem in Eq. (3.3). At each iteration of the stochastic primal-dual method, we compute the following deterministic gradient
(3.4) ∇λLΨ(θt,λt)=Ψθt.
Furthermore, we draw a random index , and subsequently sample the state-action . Then, we compute the stochastic gradient as follows
(3.5a) ∇θˆLΨ(θt,λt) =cat(st)Ψat(st)−λt,at(st)Ψat(st)1m∑di=1ψiat(st),
where is a row vector, corresponding to the row of the feature matrix . We notice that
is an unbiased estimator of the gradients of the Lagrangian function
. Formally,
IE[∇θˆLΨ(θt,λt)] =∑(s,a)∈S×Acat(st)Ψat(st)−λt,at(st)Ψat(st)1m∑di=1ψiat(st)IP[st=s,at=a] =∑(s,a)∈S×Aca(s)Ψa(s)−λt,a(s)Ψa(s) (3.6) =cTΨ−λTtΨ=∇θLΨ(θt,λt).
Algorithm 1 describes the primal-dual steps to update . To implicitly control the size of the policy class as characterized by the constraint , in Algorithm 1 we consider a Bregman divergence whose modulus has the domain . For example, when the policy class is defined by a Euclidean ball , the following convex function can be employed (see [15])
(3.7) ϕ(θ)=−√R2−∥θ∥22.
The associated convex conjugate of the modulus (3.7) is
(3.8) ϕ∗(θ)=R√1+∥θ∥22.
Moreover, the modulus (3.7) yields the Hellinger-like divergence
(3.9) Dϕ(θ1,θ2)=R2−⟨θ1,θ2⟩√R2−∥θ2∥22−√R2−∥θ1∥22.
Alternatively, when the policy class is defined by a hyper-cube , a function similar to the symmetric logistic function yields the divergence with the desired domain. In particular, let
(3.10) ϕ(θ)=d∑i=1[(R+θi)log(R+θi)+(R−θi)log(R−θi)].
The convex conjuagte of is the following function
(3.11) ϕ∗(θ)=−d∑i=1Rθi−d∑i=12Rlog(2Reθi+1).
The Bregman divergence associated with the modulus in Eq. (3.10) is
(3.12) Dϕ(θ1,θ2)=d∑i=1[(θi1+R)log(θi1+R)(θi2+R)+(R−θi1)log(R−θi1)(R−θi2)].
The steps (3.25a)-(3.25c) of Algorithm 1 are reminiscent of the so-called Mirror Descent Algorithm for online convex optimization [16],
(3.13)
However, the role of the Bregman divergence in Algorithm 1 is different from that of the Mirror Descent algorithm. In the Mirror Descent algorithm, the Bregman divergence is typically used to adapt to the geometry of the feasible set and achieve better dependency on the dimension of the embedding space of the feasible set. In contrast, the Bregman divergence in Algorithm 1 is employed to implicitly enforce the constraints due to the size of the policy class and eliminate the projection step.
To see this equivalence, first notice that the update rule in Eq. (3.13) can be rewritten as below
(3.14a) ˜θt =∇ϕ∗(∇ϕ(θt)−ηt∇θˆLΨ(θt,λt)), (3.14b) θt+1 =argminθ∈Θ2Dϕ(θ;˜θt),
where we also recall . Now, consider the Bregman projection (3.14b) onto the hyper-plane . As shown in [15], the Bregman projection in Equation (3.14b) can be alternatively computed using Eqs. (3.25b)-(3.25c). To demonstrate this, first invoke the necessary and sufficient KKT conditions for the optimality of which requires the existence of an optimal Lagrange multiplier that satisfies the following equation
(3.15) ∇θDϕ(θ;˜θt)|θ=θt+1=zt∇θ(θT1−1).
From Eq. (3.15), we establish that
(3.16) ∇ϕ(θt+1)=zt1+∇ϕ(˜θt).
Alternatively, since the gradient of a Legendre function is a bijection from to and its inverse is the gradient of the conjugate (cf. [17, Thm. 26.5]), we have
(3.17) θt+1=∇ϕ
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# BBO Discussion Forums: How will this hands be bid ? - BBO Discussion Forums
• 2 Pages
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## How will this hands be bid ?
### #21mtn1ms
• Group: Members
• Posts: 2
• Joined: 2019-April-16
Posted 2019-April-16, 19:42
Our 1 shows 5+ or 444 with 1 so we would open this 1 and respond 3 (showing limit raise in ). The bid would then go 3 (cue - A or K or singleton or void) 3 (cue - A or K or singleton or void) showing interest in 3N or 5. Opener would then bid 4 (cue - A or K or singleton or void) and responder would cue 4. Opener would cue 4 and with his 2 doubletons, knowing we have 1st and 2nd round control of and and a control in responder would bid 6. Without the A responder would bid 4 and opener would bid 5.
So our bidding would be:
1-3
3-3
4-4
4-6
Without interest or without an available cue over 3 responder could bid 5 or 3N with suitable hand.
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### #22mtn1ms
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Posted 2019-April-16, 19:42
Our 1 shows 5+ or 444 with 1 so we would open this 1 and respond 3 (showing limit raise in ). The bid would then go 3 (cue - A or K or singleton or void) 3 (cue - A or K or singleton or void) showing interest in 3N or 5. Opener would then bid 4 (cue - A or K or singleton or void) and responder would cue 4. Opener would cue 4 and with his 2 doubletons, knowing we have 1st and 2nd round control of and and a control in responder would bid 6. Without the A responder would bid 4 and opener would bid 5.
So our bidding would be:
1-3
3-3
4-4
4-6
Without interest or without an available cue over 3 responder could bid 5 or 3N with suitable hand.
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### #23GrahamJson
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• Joined: 2014-October-11
Posted Yesterday, 11:32
With my regular partner we play a jump to 3NT on the second round as showing a shortage in responder’s suit, so the bidding could go 1D-2C-3NT. It might then continue 4D- cue bids...-6D. (The 3NT bid also shows a good suit and good values).
On the other hand it’s a borderline 2C bid and if responder bids 1NT all roads lead to 3NT.
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### #24mrt2000
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• Joined: 2010-October-30
Posted Yesterday, 12:37
nige1, on 2019-April-15, 11:55, said:
MsJeniffer's deal is of the kind that authorities might flag in an effort to diagnose and prosecute collusive cheating.
Bidding double-dummy, cheats might well reach 6
Admittedly, rarely, a lucky innocent pair might also bid and make the slam.
But evidence of consistent success on such flagged deals, would corroborate cheating suspicions.
I have seldom heard such nonsense in all my life. Although I admit that both hands have to take very optimistic views of their holdings it is a million miles away from anything even remotely close to cheating. My partner of choice and I, playing 2/1 GF, could bid the hands as follows:
1 - 1NT
3 - 3 (Cue bid agreeing )
3 (Q) - 4 (No control)
4 (Q) - 4 (Q)
6 - P
Not so hard really was it?
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### #25Cyberyeti
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• Joined: 2009-July-13
• Location:England
Posted Yesterday, 13:16
mrt2000, on 2019-April-17, 12:37, said:
I have seldom heard such nonsense in all my life. Although I admit that both hands have to take very optimistic views of their holdings it is a million miles away from anything even remotely close to cheating. My partner of choice and I, playing 2/1 GF, could bid the hands as follows:
1 - 1NT
3 - 3 (Cue bid agreeing )
3 (Q) - 4 (No control)
4 (Q) - 4 (Q)
6 - P
Not so hard really was it?
It's hard because at MPs if partner doesn't have a heart cue or has 2 losing or potentially losing clubs, you may struggle to get out in a number of NT you can make, and end up playing 5 for a bottom once you blast past 3N. There are lots of variances from the actual hands which make the slam bad with the same auction
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### #26hrothgar
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Posted Yesterday, 13:53
mrt2000, on 2019-April-17, 12:37, said:
I have seldom heard such nonsense in all my life. Although I admit that both hands have to take very optimistic views of their holdings it is a million miles away from anything even remotely close to cheating. My partner of choice and I, playing 2/1 GF, could bid the hands as follows:
1 - 1NT
3 - 3 (Cue bid agreeing )
3 (Q) - 4 (No control)
4 (Q) - 4 (Q)
6 - P
Not so hard really was it?
How would your auction differ if North had one more card in either major?
Alderaan delenda est
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### #27johnu
• Posts: 2,581
• Joined: 2008-September-10
Posted Yesterday, 14:01
mrt2000, on 2019-April-17, 12:37, said:
I have seldom heard such nonsense in all my life. Although I admit that both hands have to take very optimistic views of their holdings it is a million miles away from anything even remotely close to cheating. My partner of choice and I, playing 2/1 GF, could bid the hands as follows:
1 - 1NT
3 - 3 (Cue bid agreeing )
3 (Q) - 4 (No control)
4 (Q) - 4 (Q)
6 - P
Not so hard really was it?
Great bidding looking at both hands. If I look at all 4 hands, I can also bid to contracts that depend on very lucky breaks, and avoid contracts that go down on bad or awful breaks.
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### #28nullve
• Posts: 1,197
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• Gender:Male
• Location:Norway
• Interests:partscores
Posted Yesterday, 14:03
mrt2000, on 2019-April-17, 12:37, said:
I have seldom heard such nonsense in all my life. Although I admit that both hands have to take very optimistic views of their holdings it is a million miles away from anything even remotely close to cheating. My partner of choice and I, playing 2/1 GF, could bid the hands as follows:
1 - 1NT
3 - 3 (Cue bid agreeing )
3 (Q) - 4 (No control)
4 (Q) - 4 (Q)
6 - P
Not so hard really was it?
No, it's not so hard if opps' silence convinces North that South has 0-1 clubs and not e.g. Kx. But North might feel more confident about that after
1-1N (NAT unBAL or 20-22 BAL \\ 0-12, NAT)
2-2 ("16-18"*, any \\ GF relay)
2N-3 (4+ H or 1-suited \\ relay)
3-.... (1-suited \\ ---)
in my system: Here are are 100 random deals consistent with South's bidding (modulo finer points of hand evaluation) and North having the above hand:
Spoiler
Now take a look at the EW cards on the deals where South has 2-3 clubs. Don't you think opps would interfere over 1 or 1N on the vast majority of those deals?
Btw, I believe my auction at IMPs would continue
....-3 (--- \\ relay)
4-4 (3361 \\ key card ask with diamonds agreed)
4-4N (even # of key cards \\ trump Q ask)
5N-6 (trump Q, K, K, no Q \\ contract)
P.
It's harder at MPs, of course, since belly-landing in 5 after e.g.
....-4N (--- \\ trump Q ask)
5-.... (no trump Q \\ ---)
is not really an option.
* or more precisely: meets the rule of 25 (like the unbalanced part of a Precision 1 opening), but not the rule of 28
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### #29Lorneg
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• Joined: 2011-February-21
Posted Yesterday, 14:49
test
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### #30rmnka447
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• Joined: 2012-March-18
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• Location:Illinois
• Interests:Bridge, Golf, Soccer
Posted Yesterday, 15:01
Playing 2/1 with strong NT, 5cM, I think
1 - 1 NT
3 - 3 NT
would be most probable.
Playing K-S (2/1, weak NT, 5cM), after 1 , responder has a choice of 2 bids -- 2 C which isn't a GF or 2 NT. 1 NT is defined as only 5-8 in this system to provide an escape with a poor hand with no 4cM opposite a "strong NT" type hand. Whichever response is chosen again probably leads to 3 NT at MPS.
Probing for a slam that makes because of a perfect mesh of the hands might be an academic exercise, but probably isn't going to happen in the real world.
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# Tag Info
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Your lecturer is trying to express some useful intuition, but in a vague way. To make it as simple as possible, let's go back to the simple harmonic oscillator. Recall that $$[a, a^\dagger] = 1, \quad [H, a^\dagger] = a^\dagger$$ and let's work throughout with units $k = m = 1$. Now suppose that some state has energy $n$, $$H |\psi\rangle = n |\psi \rangle.... 7 Let me put on my math hat by providing a counterexample: Let's say$$ A = a_1, \\B= a_2, \\C= a_3, \\D = a_4 $$where a_i are regular bosonic anhiliation operators, therefore$$ [a_i, a_j] \equiv 0 $$If OP's proposition works:$$ \{ A B , \, C D \} \equiv A B C D + C D A B = 2a_1a_2a_3a_4 $$will be reduced to zero, since any combination of [a_i, a_j] ... 5 If you want to be formal, the function \psi : \mathbb{R}\to \mathcal{H}, t\mapsto \lvert\psi(t)\rangle needs to be understood as a function between Banach spaces (every Hilbert space is in particular a Banach space). The correct notion of derivative is then the Fréchet derivative. Note that this vector-valued function is much easier to differentiate ... 5 You are asking about the "physical meaning" of the celebrated Weierstrass transform, which is used routinely in physics, of course:$$ \bbox[yellow]{ e^{\partial_x^2}f(x) =\frac{1}{\sqrt{4\pi}} \int_{-\infty}^\infty f(x-y)~ e^{-y^2/4}\;dy}~. $$In your case,$$\langle x |\exp(-a {\hat{p}}^2)| \psi \rangle = \exp(a \hbar^2 \partial_x^2)~\langle x|\psi \...
5
Yes it acts on the $A_x$: $$(-i\hbar \partial_x A_x) \psi= ( -i\hbar A_x\partial_x) \psi+ (-i\hbar \psi \partial_x) A$$
4
$\hat{D}_x(x)=e^{-i\frac{x}{\hbar}\hat{p}_x}$: the spatial displacement operator, moves the wave function $\psi$ along the x coordinate, $\hat{p_{x}}$ is the momentum operator which generates the displacement. First let me say that using $x$ in the definition of $\hat{D}_x(x)$ is really confusing, because $x$ is also used as the spatial coordinate of ...
4
You are indeed missing some pieces. You can immediately see that there is no way your formula works in general since the full expansion reads \begin{aligned} e^{ix\cdot P} K_\mu e^{-ix\cdot P} &= \sum_{n,m=0}^\infty \frac{i^{n-m}}{n!m!} (x\cdot P)^n\,K_\mu\,(x\cdot P)^m\,. \end{aligned}\tag{1}\label{ini} The problem is that this is a mess because ...
4
It is just how the operators work in the position basis. I will show how it works for the position operator. The momentum operator can be handled similarly First, we know that matrix elements of the position operator in the position basis are given by $\langle x|\hat X|x'\rangle=x'\delta(x'-x)$. Second, we know the position basis vectors form a complete ...
4
All of my work is based on the assumption that I can write the following:$$\Pi|\vec{r}\rangle = -|\vec{r}\rangle$$ This is wrong. $|{-\vec r}\rangle$ is an eigenvector of the position operator $\hat{\vec x}$ with eigenvalue $-\vec r$. $-|{\vec r}\rangle$ is an eigenvector of the position operator $\hat{\vec x}$ with eigenvalue $+\vec r$, since $$\hat{\vec ... 3 Yes, indeed, you simply need to calculate the matrix elements of the Hamiltonian in the new basis: \begin{array} \hat{H}_{11}' = \langle \phi_1'|\hat{H}|\phi_1'\rangle = \frac{1}{2}(\langle\phi_1| + \langle\phi_2|)\hat{H}(|\phi_1\rangle + |\phi_2\rangle) = \\ \frac{1}{2}(\langle \phi_1|\hat{H}|\phi_1\rangle + \langle \phi_1|\hat{H}|\phi_2\rangle + \langle \... 3 Okay, so I cleaned up the latex a bit (Use \langle and \rangle for the braket notation. Though I wish they'd install the braket package, which would make it even easier). First, you have a mistake in the first line, and it doesn't quite make sense when going to second quantization anyway: There should be no sum over individual particles any more! In your ... 3 If you had many systems in the state |\alpha\rangle and you were to make measurements of the photon number for each of those systems, then \langle\alpha|\hat n|\alpha\rangle is the expectation value of those measurements. This can be calculated using inner products if you can directly calculate \hat n|\alpha\rangle and \langle\alpha|\left(\hat n|\... 3 The previous answer gives the meaning, here is the calculation. The coherent state \mid \alpha\rangle is an eigenvectior of the annihilation operator \mathsf{a}, with$$\mathsf{a}\mid \alpha\rangle= \alpha \mid \alpha\rangle .$$The Hermitian conjugate of this equation is:$$\big(\mathsf{a}\mid \alpha\rangle\big)^+ = \langle\alpha\mid\mathsf{a}^+ = \...
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A nice way to see what's happening is to construct an approximate "toy" state $\newcommand{\ket}[1]{\left|#1\right>} \ket r$. For instance you might construct a wavefunction that's "at" a location ten units to the right of your origin by $$\ket{10}=\left\{ \begin{array}{cl} 1 & \text{where } 9<x<11 \\ 0 & \text{elsewhere} \end{array} \... 3 Like all proofs about properties of "maximal" sets in any context, this one too proceeds by assuming that we have a set that lacks the property and then constructing something we can add to the set, showing it was not maximal: Assume we have a set of n operators A_i and that there is a degenerate common eigenvector, i.e. an (w.l.o.g.) two -dimensional ... 2 Expressing the momentum operator as spatial derivative gives us:$$ \hat D_z=\exp\left(-i\frac{z}{\hbar}(-i\hbar\partial_x)\right)=\exp(z\partial_x) $$The result of \left.\hat D_z\psi\middle|x\right> can be expressed as:$$ \left.\hat D_z\psi\middle|x\right> = \psi(x)+z\psi'(x)+\frac{z^2}2\psi''(x)+\frac{z^3}6\psi'''(x)\ldots $$which can be seen ... 2 In order for your formula to be valid, the states |i\rangle must be eigenvectors of H with the energy e_i. Otherwise you will not get$$\langle j | G_r | i \rangle = \langle j| \frac{1}{E_f-e_i+i\gamma} |i\rangle$$as what you did was explicitly act with H on the state to the right. While |j\rangle might be any basis of states whatsoever, it will ... 2 The trace of any matrix/operator will be same regardless of what basis you use, provided they are complete. So it doesn't matter whether you choose eigenvectors of A or H, but you must be consistent and use the complete basis. If you decide to use the eigenbasis of A, then you can't simply substitute the scalar energy E_i for H, you must keep H ... 2 Because mass is supposed to be a quantity which is to remain constant for all eternity in quantum mechanics. Mass is energy in the rest frame of the of the particle. The reason why you don’t see this in non-relativistic quantum mechanics is that you can add an arbitrary constant to the Hamiltonian. In relativistic quantum mechanics, that offset is fixed with ... 2 You have$$\sum_l \bar U_{lj} a_l = \sum_l U^\dagger_{jl} a_ l \neq \sum_l \big(U^\dagger_{jl} a_l\big)^\dagger$$and if you had started from \{\bar a_i, \bar a_j^\dagger\} , you'd have$$ \{\bar a_i, \bar a_j^\dagger\} = \big\{\sum_k \bar U_{ki} a_k, (\sum_l \bar U_{lj} a_l)^\dagger\big\} = \sum_k \sum_l \big\{\bar U_{ki} a_k, (\bar U_{lj} a_l)^\dagger\...
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The conditions, $[A,[A,B]]=[B,[A,B]]=0$, are needed for commuting operators to satisfy the identity that $[A,F(B)]=[A,B]\frac{\partial F}{\partial B}$. It turns out that these conditions are not needed if one works with anticommuting operators A and B (i.e., $\{A,A\}=\{B,B\}=0$) instead. To show this, we note that any function $F(B)$ constructed from an ...
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The easiest way to derive your solution is probably to convert the first equality into a differential equation: $$a(t)=e^{iHt}ae^{-iHt}\qquad\Longrightarrow\qquad \frac{d}{dt}a(t)=e^{iHt}(iHa-aiH)e^{-iHt}=ie^{iHt}[H,a]e^{-iHt}$$ where I used $$\frac{d}{dt}e^{iHt}=iHe^{iHt}=e^{iHt}iH\qquad\qquad \frac{d}{dt}e^{-iHt}=-iHe^{-iHt}=e^{iHt}(-iH)$$ It ...
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A Hilbert space is, by definition, an inner product space (meaning that it is also a normed space) and it is Cauchy Complete. Completeness means that all the normal definitions of limiting procedures go through as usual (with no more than minor notational changes).
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Short answer: Intuitively, for the displacement operator, the exponential accumulates an infinite number of infinitesimal displacements, and this gives rise to an overall macroscopic finite displacement. The same principle holds for the rotation operator, i.e., accumulation of many small rotations. Since the momentum operator generates a displacement via ...
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First of all, the phrasing of the question might be a bit misleading, since you are obviously talking not about any state, but about any eigenstate of $\mathbf{\hat{A}}$, since it satisfies the property $$\mathbf{\hat{A}}|\psi\rangle = a|\psi\rangle.$$ Note, btw, that $\mathbf{\hat{A}}$ is not necessarily Hermitian (otherwise $a$ ...
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This is not an assumption, it is a requirement for consistency. The symmetry transformation acts on operators and states, it does not act on numbers. So the equation $A\lvert \psi_n \rangle = a_n\lvert \psi_n\rangle$ simply becomes $A'\lvert \psi_n'\rangle = a_n\lvert \psi_n'\rangle$ after applying the transformation. This equation must be true for any ...
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It is a different application of the word determinate. Griffiths simply means that the result of a measurement is determinate if the state is an eigenfunction. That is not the same as determinism in time evolution.
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In the Schrödinger picture, the Hilbert space $\mathcal{H}$ is physically the set of states at a given time. A function like $\psi(x,t)$ is not a state, but a time evolution of a state. Operators are also a priori not time dependent: they take functions of $x$ and return functions of $x$. A time dependent operator is really an operator valued function; you ...
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A physical way to think about this is that, if you have a degeneracy then you have a symmetry. Namely, if $|1\rangle,|2\rangle$ are states with the same set of eigenvalues $\lambda_1,\ldots,\lambda_n$ under $A_1,\ldots,A_n$, then there exists an operator that rotates $|1\rangle$ and $|2\rangle$ into each other and leaves all other observables untouched $$U(\... 1 I'll write down the whole calculation for completion. Firstly we define$$A = C-\langle C\rangle \qquad B = D-\langle D \rangle$$We firstly evaluate the following$$\frac{\langle v|A^2|v\rangle}{\langle v | v \rangle} = \frac{\langle v|C^2-2C\langle C\rangle+\langle C\rangle^2|v\rangle}{\langle v | v \rangle} = \frac{\langle v|C^2|v\rangle}{\langle v | v ...
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# American Institute of Mathematical Sciences
November 2020, 14(4): 603-611. doi: 10.3934/amc.2020033
## Designs from maximal subgroups and conjugacy classes of Ree groups
1 School of Mathematical Sciences, North-West University, (Mafikeng) 2754, South Africa 2 School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal, Durban 4000, South Africa
* Corresponding author: Seiran Zandi
Received August 2018 Revised August 2019 Published November 2019
Fund Project: The first author acknowledges support of NRF and NWU (Mafikeng).
The second author acknowledges support of NRF through Grant Numbers 95725 and 106071.
The third author acknowledges support of NWU (Mafikeng) postdoctoral fellowship.
The fourth author acknowledges support of NRF postdoctoral fellowship through Grant Number 91495
In this paper, using a method of construction of $1$-designs which are not necessarily symmetric, introduced by Key and Moori in [5], we determine a number of $1$-designs with interesting parameters from the maximal subgroups and the conjugacy classes of the small Ree groups $^2G_2(q)$. The designs we obtain are invariant under the action of the groups $^2G_2(q)$.
Citation: Jamshid Moori, Bernardo G. Rodrigues, Amin Saeidi, Seiran Zandi. Designs from maximal subgroups and conjugacy classes of Ree groups. Advances in Mathematics of Communications, 2020, 14 (4) : 603-611. doi: 10.3934/amc.2020033
##### References:
show all references
##### References:
Non-trivial designs from $G = Ree(q)$ using construction Method 2
$Max$ $t =o(x)$ $v = |x^G|$ $k =| M \cap x^G|$ $\lambda= \chi_{M_i}(x)$ $M_1$ $t=2$ $q^2(q^2-q+1)$ $q^2$ $q+1$ $M_1$ $t=3$ $(q^3+1)(q-1)$ $q-1$ 1 $M_1$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q(q-1)}{2}$ 1 $M_1$ $t=9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q^2(q-1)}{3}$ 1 $M_1$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q^2 (q-1)}{2}$ 1 $M_1$ $t |(q-1)$, $t \ne 2$ ${q^3}(q^3+1)$ $2q^3$ 2 $M_2, M_3$ $t=2$ $q^2(q^2-q+1)$ $q^{\mp}$ $\frac{{{q(q^2-1)}}}{6}$ $M_2, M_3$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $q^{\mp}$ $\frac{q^2}{3}$ $M_2, M_3$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $q^{\mp}$ $\frac{q}{3}$ $M_2, M_3$ $t | q^{\mp}$ ${q^3(q^2-1)q^{\pm}}$ $6$ $1$ $M_4$ $t=2$ $q^2(q^2-q+1)$ $q^2-q+1$ $q^2-q+1$ $M_4$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q^2-1}{2}$ $q$ $M_4$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q^2-1}{2}$ $1$ $M_4$ $t |(q-1)$, $t \ne 2$ $q^3(q^3+1)$ $q(q+1)$ $1$ $M_4$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $3q(q-1))$ $3$ $M_5$ $t=2$ $q^2(q^2-q+1)$ $q+4$ $\frac{{{q(q-1)(q+4)}}}{6}$ $M_5$ $t=3$ $\frac{{{{q}(q^3+1)(q-1)}}}{2}$ $q+1$ $\frac{{{q^2}}}{3}$ $M_5$ $t=6$ $\frac{{{{q^2}(q^3+1)(q-1)}}}{2}$ $q+1$ $\frac{{{q}}}{3}$ $M_5$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $6$ 1 $M_6$ $t=2$ $q^2(q^2-q+1)$ $q_0^2(q_0^2-q_0+1)$ $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ $M_6$ $t=3$ $(q^3+1)(q-1)$ $(q_0^3+1)(q_0-1)$ $\frac{{{q^3}}}{q_0^3}$ $M_6$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q_0(q_0^3+1)(q_0-1)}{2}$ $\frac{{{q^2}}}{q_0^2}$ $M_6$ $t= 9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{3}$ $\frac{{{q}}}{q_0}$ $M_6$ $t= 6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{2}$ $\frac{{{q}}}{q_0}$ $M_6$ $t |(q_0-1)$, $t \ne 2$ ${q^3(q^3+1)}$ ${q_0^3(q_0^3+1)}$ $\frac{q-1}{q_0-1}$ $M_6$ $t |\frac{q_0+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ ${q_0^3(q_0^2-q_0+1)(q_0-1)}$ $\frac{q+1}{q_0+1}$ $^*M_6$ $t|q_0^{\pm}$ ${q^3}\left( {q^2 - 1} \right){q^{\pm}}$ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}}$ $\frac{{q^{\mp}}}{{q_0^{\mp}}}$ $^{**}M_6$ $t|q_0^{\pm}$ ${q^3(q^3+1)}$ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}}$ $\frac{q-1}{q_0^{\mp}}$
$Max$ $t =o(x)$ $v = |x^G|$ $k =| M \cap x^G|$ $\lambda= \chi_{M_i}(x)$ $M_1$ $t=2$ $q^2(q^2-q+1)$ $q^2$ $q+1$ $M_1$ $t=3$ $(q^3+1)(q-1)$ $q-1$ 1 $M_1$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q(q-1)}{2}$ 1 $M_1$ $t=9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q^2(q-1)}{3}$ 1 $M_1$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q^2 (q-1)}{2}$ 1 $M_1$ $t |(q-1)$, $t \ne 2$ ${q^3}(q^3+1)$ $2q^3$ 2 $M_2, M_3$ $t=2$ $q^2(q^2-q+1)$ $q^{\mp}$ $\frac{{{q(q^2-1)}}}{6}$ $M_2, M_3$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $q^{\mp}$ $\frac{q^2}{3}$ $M_2, M_3$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $q^{\mp}$ $\frac{q}{3}$ $M_2, M_3$ $t | q^{\mp}$ ${q^3(q^2-1)q^{\pm}}$ $6$ $1$ $M_4$ $t=2$ $q^2(q^2-q+1)$ $q^2-q+1$ $q^2-q+1$ $M_4$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q^2-1}{2}$ $q$ $M_4$ $t=6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q^2-1}{2}$ $1$ $M_4$ $t |(q-1)$, $t \ne 2$ $q^3(q^3+1)$ $q(q+1)$ $1$ $M_4$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $3q(q-1))$ $3$ $M_5$ $t=2$ $q^2(q^2-q+1)$ $q+4$ $\frac{{{q(q-1)(q+4)}}}{6}$ $M_5$ $t=3$ $\frac{{{{q}(q^3+1)(q-1)}}}{2}$ $q+1$ $\frac{{{q^2}}}{3}$ $M_5$ $t=6$ $\frac{{{{q^2}(q^3+1)(q-1)}}}{2}$ $q+1$ $\frac{{{q}}}{3}$ $M_5$ $t |\frac{q+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ $6$ 1 $M_6$ $t=2$ $q^2(q^2-q+1)$ $q_0^2(q_0^2-q_0+1)$ $\frac{q(q^2-1)}{q_0(q_0^2-1)}$ $M_6$ $t=3$ $(q^3+1)(q-1)$ $(q_0^3+1)(q_0-1)$ $\frac{{{q^3}}}{q_0^3}$ $M_6$ $t=3$ $\frac{q(q^3+1)(q-1)}{2}$ $\frac{q_0(q_0^3+1)(q_0-1)}{2}$ $\frac{{{q^2}}}{q_0^2}$ $M_6$ $t= 9$ $\frac{q^2(q^3+1)(q-1)}{3}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{3}$ $\frac{{{q}}}{q_0}$ $M_6$ $t= 6$ $\frac{q^2(q^3+1)(q-1)}{2}$ $\frac{q_0^2(q_0^3+1)(q_0-1)}{2}$ $\frac{{{q}}}{q_0}$ $M_6$ $t |(q_0-1)$, $t \ne 2$ ${q^3(q^3+1)}$ ${q_0^3(q_0^3+1)}$ $\frac{q-1}{q_0-1}$ $M_6$ $t |\frac{q_0+1}{2}$, $t \ne 2$ ${q^3(q^2-q+1)(q-1)}$ ${q_0^3(q_0^2-q_0+1)(q_0-1)}$ $\frac{q+1}{q_0+1}$ $^*M_6$ $t|q_0^{\pm}$ ${q^3}\left( {q^2 - 1} \right){q^{\pm}}$ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}}$ $\frac{{q^{\mp}}}{{q_0^{\mp}}}$ $^{**}M_6$ $t|q_0^{\pm}$ ${q^3(q^3+1)}$ ${q_0^3}\left( {q_0^2 - 1} \right){q_0^{\pm}}$ $\frac{q-1}{q_0^{\mp}}$
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# Applied Geometry
0
358
2
find the circumfence of a circle that is inscribed in a square with side 12 cm.
Aug 16, 2017
### Best Answer
#1
+349
+1
If the side of the square is 12 cm, then the radius of the circle is half of that, or 6 cm. The circumference of a circle is 2*Pi*r. Substituting r, we get
$$2\Pi6cm=12\Pi\approx37.7cm$$
.
Aug 16, 2017
### 2+0 Answers
#1
+349
+1
Best Answer
If the side of the square is 12 cm, then the radius of the circle is half of that, or 6 cm. The circumference of a circle is 2*Pi*r. Substituting r, we get
$$2\Pi6cm=12\Pi\approx37.7cm$$
Mathhemathh Aug 16, 2017
#2
+2298
0
That was a creative way of making the symbol for pi! For future reference, use /pi in Latex to get the desired symbol. It looks like this:
$$\pi$$
TheXSquaredFactor Aug 16, 2017
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Research
# Suzuki-type fixed point results in metric type spaces
Nawab Hussain1*, Dragan Ðorić2, Zoran Kadelburg3 and Stojan Radenović4
Author Affiliations
1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia
2 Faculty of Organizational Sciences, University of Belgrade, Jove Ilića 154, Beograd, 11000, Serbia
3 Faculty of Mathematics, University of Belgrade, Studentski trg 16, Beograd, 11000, Serbia
4 Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Beograd, 11120, Serbia
For all author emails, please log on.
Fixed Point Theory and Applications 2012, 2012:126 doi:10.1186/1687-1812-2012-126
The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/126
Received: 5 February 2012 Accepted: 18 July 2012 Published: 31 July 2012
© 2012 Hussain et al.; licensee Springer
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
### Abstract
Suzuki’s fixed point results from (Suzuki, Proc. Am. Math. Soc. 136:1861-1869, 2008) and (Suzuki, Nonlinear Anal. 71:5313-5317, 2009) are extended to the case of metric type spaces and cone metric type spaces. Examples are given to distinguish our results from the known ones.
MSC: 47H10, 54H25.
##### Keywords:
metric type space; cone metric space; normal cone; fixed point
### 1 Introduction and preliminaries
In 2008 Suzuki proved the following refinement of Banach’s fixed point principle.
Theorem 1 ([1], Theorem 2])
Letbe a complete metric space. Letbe a selfmap andbe defined by
(1.1)
If there existssuch that for each,
thenThas a unique fixed pointand for each, the sequenceconverges to z.
There were various extensions of Suzuki’s result, such as Kikkawa-Suzuki’s version of Kannan’s theorem [2] and Popescu’s version of Ćirić’s theorem [3].
Suzuki proved also the following version of Edelstein’s fixed point theorem.
Theorem 2 ([4], Theorem 3])
Letbe a compact metric space. Letbe a selfmap, satisfying for all, the condition
ThenThas a unique fixed point inX.
This theorem was generalized in [5].
Let E be a real Banach space with the zero vector θ. A subset P of E is called a cone if: (a) P is closed, non-empty and ; (b) , , imply that ; (c) . Given a cone P, we define the partial ordering ⪯ with respect to P by if and only if . We shall write for , where intP stands for the interior of P and use for and . If , then P is called a solid cone. It is said to be normal if there is a number such that for all , implies . Such a minimal constant K is called the normal constant of P.
Huang and Zhang re-introduced cone metric spaces in [6] (this notion was known under various names since the mid of the 20th century, see a survey in [7]), replacing the set of real numbers by an ordered Banach space as the codomain for a metric. Cone metric spaces over normal cones inspired another generalization of metric spaces that were called metric type spaces by Khamsi [8] (see also [9-12]; note that, in fact, spaces of this kind were used earlier under the name of b-spaces by Czerwik [13]). Cvetković et al.[14] and Shah et al.[15] extended Khamsi’s definition and defined cone metric type spaces as follows:
Definition 1 ([14,15])
Let X be a nonempty set, E a Banach space with the solid cone P and let be a real number. If the function satisfies the following properties:
(a) if and only if ;
(b) for all ;
(c) for all ,
then D is called a cone metric type function and is called a cone metric type space (CMTS).
In particular, when and , CMTS reduces to a metric type space (MTS) of [8,9,12].
Of course, for we get the cone metric space (CMS) of [6], resp. the usual metric space.
Example 1 ([14])
Let be an orthonormal basis of with inner product and let . Define
where is the class of functions being equal to the function f a.e. Further, let
and let be defined by
It was shown in [14] that is a solid cone in and that is a CMTS. In particular, for we get an MTS and for a CMS.
Example 2 ([8,10])
Let be any CMS over a normal cone with normal constant . Then is an MTS, where . In this case the spaces and have the same topologies (see [10], Theorem 2.7]).
If is a CMTS over a normal cone with a normal constant , then is an MTS, where . Similarly as above, the spaces and have the same topologies.
Notions such as convergent and Cauchy sequences, as well as completeness, are introduced in (cone) metric type spaces in the standard way. The following obviously holds in an arbitrary (cone) metric type space:
We will sometimes need the continuity of metric-type function D in one variable:
or in two variables:
The last property always holds in the case of an MTS generated by a CMS over a normal cone, see Example 2, but not in general, as the following example shows.
Example 3 Let and let be defined by
Then it is easy to see that for all , we have
Thus, is a metric-type space. Let for each . Then
that is, , but as .
Recall that a selfmap is said to have the property (P) [16] if for each , where is the set of fixed points of T.
In this paper, we extend Suzuki’s Theorems 1 and 2, as well as Popescu’s results from [3] to the case of metric type spaces and cone metric type spaces. Examples are given to distinguish our results from the known ones.
### 2 Results
#### 2.1 Results in metric type spaces
Theorem 3Letbe a complete MTS whereDis continuous in each variable. Letbe a selfmap andbe defined by
(2.1)
whereis the positive solution of. If there existssuch that for each,
(2.2)
where
thenThas a unique fixed pointand for each, the sequenceconverges to z. Moreover, Thas the property (P).
Note that for , Theorem 3 reduces to a special case of Theorem 2.1 by Popescu [3].
Proof First note that implies that and it follows by (2.2) that
wherefrom
(2.3)
for each .
Let be arbitrary and form the sequence by and for . It follows from (2.3) that
(2.4)
and, by induction,
(2.5)
Using [12], Lemma 3.1] we conclude that is a Cauchy sequence, tending to some z in the complete space X. Obviously, also .
Let us prove now that
(2.6)
holds for each . Since and (and hence ) and, by continuity of D, , it follows that there exists such that
holds for each . Assumption (2.2) implies that for such n
Passing to the limit when (and using continuity of D), we get that
It is easy to see that (2.6) follows from the previous relation.
Putting in (2.3), we get that
(2.7)
holds for each (where ). It follows by induction that
(2.8)
We will prove now that
(2.9)
for each . For this relation is obvious. Suppose that it holds for some . If , then and . If , then we can apply (2.6) to obtain that
Using (2.8) and the induction hypothesis, we get that
and (2.9) is proved by induction.
In order to prove that , we suppose that and consider the two possible cases.
Case I. (and hence ). We will prove first that
(2.10)
for . For this is obvious and for it follows from (2.8). Suppose that (2.10) holds for some . Then
wherefrom . It follows (using (2.8)) that
Assumption (2.2) implies that
It is easy to see (using (2.8), (2.9) and the inductive hypothesis) that the last maximum is equal to , i.e., and relation (2.10) is proved by induction.
Now and (2.10) implies that for each . Hence, (2.6) and (2.8) imply that
(2.11)
Since , it follows from (2.10) that
There exists such that for and . For such n, we have that
It follows from (2.11) that
Thus, and, again from (2.10), we get that and , a contradiction.
Case II. (and so ). We will prove that there exists a subsequence of such that
(2.12)
holds for each . From (2.4) we know that holds for each . Suppose that
both hold for some . Then
which is impossible. Hence one of the following holds for each n:
In particular,
holds for each . In other words, there is a subsequence of such that (2.12) holds for each . But then assumption (2.2) implies that
Passing to the limit when we get that , which is possible only if , a contradiction.
Thus, we have proved that z is a fixed point of T. The uniqueness of the fixed point follows easily from (2.6). Indeed, if yz are two fixed points of T, then (2.6) implies that
wherefrom . The property (P) follows from (2.3) (see [16]). □
Suzuki-Banach-type and Suzuki-Kannan-type fixed point results in metric type spaces (versions of [1], Theorem 2] and [2], Theorem 2.2]) are special cases of Theorem 3.
Corollary 1Letbe a complete MTS whereDis continuous in each variable. Letbe a selfmap andbe defined by (2.1). If there existssuch that for each,
thenThas a unique fixed pointand for each, the sequenceconverges to z. Moreover, Thas the property (P).
Corollary 2Letbe a complete MTS whereDis continuous in each variable. Letbe a selfmap andbe defined by (2.1). If there existssuch that for each,
thenThas a unique fixed pointand for each, the sequenceconverges to z. Moreover, Thas the property (P).
Corollary 3Letbe a complete MTS whereDis continuous in each variable. Letbe a selfmap andbe defined by (2.1). If there existssuch that for each,
thenThas a unique fixed pointand for each, the sequenceconverges to z. Moreover, Thas the property (P).
Adapting [1], Example 1] we give now an example of a mapping satisfying the conditions of Theorem 3 (and having a unique fixed point) but not satisfying the respective classical (non-Suzuki-type) condition in metric type spaces (see, e.g., [14], Theorem 3.4]).
Example 4 Let , and let be given by . Then is a metric type space (see Example 1). Let be given as
We will check that condition (2.2) holds true for and all . If or if , it is trivially satisfied. Let and . Then and for and for or . Hence, in any case,
Let now , . Then and and so , and (2.2) is trivially satisfied. Note that in the classical variant, in this case and , so the inequality does not hold for any .
The following is a metric-type version of Theorem 2.
Theorem 4Letbe a compact MTS, where the functionDis continuous. Letbe a selfmap, satisfying for all, the condition
(2.13)
ThenThas a unique fixed point inX.
Proof Denote and choose a sequence in X such that (). Since the space X is (sequentially) compact, we can suppose that there exist such that and (). We will prove that .
Suppose that and note that continuity of D implies that . Choose such that for all
holds true. Then and assumption (2.13) implies that for . Passing to the limit, we obtain that . If , the last inequality is impossible by the definition of β. If , it is possible only if (recall that we have supposed that ). But in this case and (2.13) implies that , which is again impossible by the definition of β. Hence, in all cases we obtain a contradiction and it follows that and so .
In order to prove that T has a fixed point, suppose that for all . Then, in particular, and (2.13) implies that
It follows that
when . Hence, ().
Suppose now that
both hold for some . Then
which is impossible. Thus, for each , either
holds true. Assumption (2.13) implies that for each either
holds. In other words, there exists a sequence such that holds for each , or there exists a sequence such that holds for each . In both cases, passing to the limit, we obtain that , i.e., , a contradiction with the assumption that T has no fixed points.
It follows that there exists such that . Uniqueness follows easily. □
#### 2.2 Results in cone metric type spaces
In this subsection, we formulate cone-metric-type versions of the results from the previous subsection.
Theorem 5Letbe a complete CMTS with the normal underlying coneP, whereis continuous in each variable. Letbe a selfmap andbe defined by (2.1). If there existssuch that for each,
(2.14)
for some
thenThas a unique fixed pointand for each, the sequenceconverges to z.
Proof Since the cone P is normal, without loss of generality, we can assume that the normal constant of P is and that the given norm in E is monotone, i.e. (see [17], Lemma 2.1]). Denote . Then D is a (real-valued) metric-type function and the space is compact (together with , see [10], Theorem 2.7]). Let us prove that the mapping T satisfies for some the condition
(2.15)
of Theorem 3. Suppose that . Then (indeed, if, to the contrary, i.e., it would follow that , a contradiction with the assumption). Assumption (2.14) implies that for some
Again by the monotonicity of the norm, this means that , where
Hence, condition (2.15) is satisfied, and the conclusion follows. □
In a similar way, the following corollaries and the theorem can be proved.
Corollary 4Letbe a complete CMTS whereis continuous in each variable. Letbe a selfmap andbe defined by (2.1). If there existssuch that for each,
thenThas a unique fixed pointand for each, the sequenceconverges to z.
Corollary 5Letbe a complete CMTS whereis continuous in each variable. Letbe a selfmap andbe defined by (2.1). If there existssuch that for each,
where, thenThas a unique fixed pointand for each, the sequenceconverges toz.
Corollary 6Letbe a complete CMTS whereis continuous in each variable. Letbe a selfmap andbe defined by (2.1). If there existssuch that for each,
thenThas a unique fixed pointand for each, the sequenceconverges to z.
Example 4 can be easily adapted to a CMTS.
Theorem 6Letbe a compact CMTS, where the functionis continuous. Letbe a selfmap satisfying, for all, the condition
(2.16)
ThenThas a unique fixed point inX.
Note that for the above theorem reduces to [5], Theorem 3.8].
### Competing interests
The authors declare that they have no competing interests.
### Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
### Acknowledgements
The first author gratefully acknowledges the support provided by the Deanship of Scientific Research (DSR), King Abdulaziz University during this research. The second, third and fourth authors are thankful to the Ministry of Science and Technological Development of Serbia.
### References
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3. Popescu, O: Two fixed point theorems for generalized contractions with constants in complete metric spaces. Cent. Eur. J. Math.. 7, 529–538 (2009). Publisher Full Text
4. Suzuki, T: A new type of fixed point theorem in metric spaces. Nonlinear Anal.. 71, 5313–5317 (2009). Publisher Full Text
5. Ðorić, D, Kadelburg, Z, Radenović, S: Edelstein-Suzuki-type fixed point results in metric and abstract metric spaces. Nonlinear Anal.. 75, 1927–1932 (2012). Publisher Full Text
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7. Zabrejko, PP: K-metric and K-normed linear spaces: survey. Collect. Math.. 48, 825–859 (1997)
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9. Khamsi, MA, Hussain, N: KKM mappings in metric type spaces. Nonlinear Anal.. 73, 3123–3129 (2010). Publisher Full Text
10. Radenović, S, Kadelburg, Z: Quasi-contractions on symmetric and cone symmetric spaces. Banach J. Math. Anal.. 5, 38–50 (2011)
11. Hussain, N, Shah, MH: KKM mappings in cone b-metric spaces. Comput. Math. Appl.. 62, 1677–1684 (2011). Publisher Full Text
12. Jovanović, M, Kadelburg, Z, Radenović, S: Common fixed point results in metric type spaces. Fixed Point Theory Appl.. 2011, (2011)
13. Czerwik, S: Contraction mappings in b-metric spaces. Acta Math. Inform. Univ. Ostrav.. 1, 5–11 (1993)
14. Cvetković, AS, Stanić, MP, Dimitrijević, S, Simić, Su: Common fixed point theorems for four mappings on cone metric type space. Fixed Point Theory Appl.. 2011, (2011)
15. Shah, MH, Simić, S, Hussain, N, Sretenović, A, Radenović, S: Common fixed point theorems for occasionally weakly compatible pairs on cone metric type spaces. J. Comput. Anal. Appl.. 14, 290–297 (2012)
16. Jeong, GS, Rhoades, BE: Maps for which . Fixed Point Theory Appl.. 6, 87–131 (2005)
17. Farajzadeh, AP, Amini-Harandi, A, Baleanu, D: Fixed point theory for generalized contractions in cone metric spaces. Commun. Nonlinear Sci. Numer. Simul.. 17, 708–712 (2012). Publisher Full Text
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# Motivation
When I use math mode inside of itemize environments for multiline equations, I have to do something like the following
...
\begin{itemize}
\item \begin{aligned}[t] equation stuff \end{aligned}
...
\end{itemize}
...
# Question
How can I wrap the two components \begin{aligned}[t] and \end{aligned} into a new symbol, e.g. §? I'd like to write the above example as
...
\begin{itemize}
\item §equation stuff§
...
\end{itemize}
...
It is favorable for the solution to be robust.
(It is not required for the new symbol to be §.)
• How about a macro \mleq defined as \newcommand\mleq[1]{\begin{aligned}[t] #1 \end{aligned}? – Herr K. Aug 21 '13 at 15:19
• – Qrrbrbirlbel Aug 21 '13 at 19:32
• @Qrrbrbirlbel Is it then not recommended to use aligned? What do you propose instead? – Henri Menke Aug 23 '13 at 13:51
• @HenriMenke No. I don’t know of a better alternative (except maybe an array solution). As you can see in the numerous linked references you will just to insert a \!. – Qrrbrbirlbel Aug 24 '13 at 5:14
The \catcode§=\active method won't work if the document declares
\usepackage[utf8]{inputenc}
because § is a two byte character in UTF-8. For this case you can use
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{newunicodechar}
\newunicodechar{§}{\mymath}
\def\mymath#1§{\begin{aligned}[t] #1\end{aligned}}
\newenvironment{itemalign}
{\aligned[t]} {\endaligned}
\begin{document}
\begin{itemize}
\item §x&=2\\y&=6§
\item \begin{itemalign}x&=2\\y&=6\end{itemalign}
\end{itemize}
\end{document}
that has the advantage of working also with LuaLaTeX and XeLaTeX (where the call to inputenc should not be present).
However, I'd recommend using the new environment method.
• You used \aligned to open the envorinment, but you used \endalignedat to close it (Note the trailing at). Anyway it works, but why? And why don't you use \endaligned? – Henri Menke Aug 23 '13 at 19:26
• @HenriMenke Thanks for noting! I checked and discovered that \endalignedat executes \endalign. ;-) But it's wrong anyway. – egreg Aug 23 '13 at 19:40
• I used the \newenvironment solution. To make it more align-like I'd like to add a new line at the end of itemalign. I used {\endaligned\par}, but I doubt, that this is the right procedure, because this will start a new paragaph after each itemalign. What do I have to insert instead of \par? – Henri Menke Aug 28 '13 at 20:56 • @HenriMenke I believe this should be the entire item, so you need nothing after it. – egreg Aug 28 '13 at 21:04 If I were you, I would use two different macros to open and close the environment. I would do something like \def\ba#1\ea{\begin{aligned}[t]#1\end{aligned}$} that can be use like that \begin{itemize} \item \ba equation stuff \ea \end{itemize} How about making § active? \documentclass{article} \usepackage{amsmath} \catcode§=\active \def§#1§{$\begin{aligned}[t] #1\end{aligned}\$}
\begin{document}
\begin{itemize}
\item §x&=2\\y&=6§
\end{itemize}
\end{document}
Then we can scan an argument up to the next occurence of § and wrap this in aligned.
As @egreg pointed out, this will be problematic using pdflatex and inputenc (see also: Catcodes of unicode characters with \usepackage[utf8]{inputenc}). A simple workaround in this case is changing § to e.g. |.
• This would not work with \usepackage[utf8]{inputenc} – egreg Aug 21 '13 at 15:44
• @egreg fair point! It works with my xelatex, however, so I didn't notice. Changing the character from § to e.g. | fixes that though. – Jonathan Aug 21 '13 at 16:22
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# ACsc Function
Returns the inverse cosecant
## Description
The Function $$ACsc$$ returns the inverse cosecant of real or complex numbers. The argument can be a single number or a data field. For data fields, the inverse cosecant of each element is calculated and the results returned in a data field of equal size.
## ACsc for real numbers
### Parameter
For real values, the range must be -infinite to -1 or 1 to infinite .
### Result
The result returns in degrees (range is -90° to +90°) or radians (range is -π/2 to +π/2). The unit of measurement is set in the toolbar with the DEG or RAD buttons. The Setting applies to the entire worksheet. Die Einstellung gilt für das gesamte Arbeitsblatt.
For arguments in the range between -1 and 1 , the result is NaN (Not a Number)
### Optional Parameter
Optionally, a second parameter with the keywords DEG or RAD can be specified to set the unit of measure for this function call. The specification of the parameter has priority over the global setting in the toolbar. For different functions, you can use different units of measurement regardless of the default setting in the toolbar.
ACsc (x)
ACsc (x, deg)
## ACsc for complex numbers
For complex numbers, the argument and the result is always given as radian, regardless of the default setting in the toolbar. The result is also a complex number.
ACsc(x + i)
### Example
ACsc(2+3i)=0.15-0.23i
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Search
# DYEING OF COTTON MATERIAL WITH DIRECT DYES (Lab Manual)
Updated: Mar 13, 2021
Aim:
To dye the given cotton material using direct dyestuff with the required percentage of dye.
Recipe:
Procedure:
The given cotton material is weighted accurately. The dye bath is prepared by pasting the dyestuff with sodium carbonate and water. The material is then steeped in the dye solution at 40o c. half the amount of sodium chloride is added after 15minutes of dyeing and the temperature of the dye solution is increased to 100o c and then the remaining amount is added and dyed for another 45 minutes. Then it is subjected in the cooling bath for obtaining better exhaustion and it is dried.
Calculation:
Result:
The given cotton sample material was dyed with direct dye.
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[Resource Topic] 2018/1217: Changing Points in APN Functions
Welcome to the resource topic for 2018/1217
Title:
Changing Points in APN Functions
Authors: Lilya Budaghyan, Claude Carlet, Tor Helleseth, Nikolay Kaleyski
Abstract:
We investigate the differential properties of a construction in which a given function F : \mathbb{F}_{2^n} \rightarrow \mathbb{F}_{2^n} is modified at K \in \mathbb{N} points in order to obtain a new function G. This is motivated by the question of determining the minimum Hamming distance between two APN functions and can be seen as a generalization of a previously studied construction in which a given function is modified at a single point. We derive necessary and sufficient conditions which the derivatives of F must satisfy for G to be APN, and use these conditions as the basis for an efficient filtering procedure for searching for APN functions whose value differs from that of a given APN function F at a given set of points. We define a quantity m_F related to F counting the number of derivatives of a given type, and derive a lower bound on the distance between an APN function F and its closest APN neighbor in terms of m_F. Furthermore, the value m_F is shown to be invariant under CCZ-equivalence and easier to compute in the case of quadratic functions. We give a formula for m_F in the case of F(x) = x^3 which allows us to express a lower bound on the distance between F(x) and the closest APN function in terms of the dimension n of the underlying field. We observe that this distance tends to infinity with n. We also compute m_F and the distance to the closest APN function for a representative F from each of the switching classes over \mathbb{F}_{2^n} for 4 \le n \le 8. For a given function F and value v, we describe an efficient method for finding all sets of points \{ u_1, u_2, \dots, u_K \} such that setting G(u_i) = F(u_i) + v and G(x) = F(x) for x \ne u_i is APN.
Feel free to post resources that are related to this paper below.
Example resources include: implementations, explanation materials, talks, slides, links to previous discussions on other websites.
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Extremal behavior of stochastic integrals driven by regularly varying L\'{e}vy processes
Abstract
We study the extremal behavior of a stochastic integral driven by a multivariate L\'{e}vy process that is regularly varying with index $\alpha>0$. For predictable integrands with a finite $(\alpha+\delta)$-moment, for some $\delta>0$, we show that the extremal behavior of the stochastic integral is due to one big jump of the driving L\'{e}vy process and we determine its limit measure associated with regular variation on the space of c\{a}dl\{a}g functions.Comment: Published at http://dx.doi.org/10.1214/009117906000000548 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org
Topics: Mathematics - Probability, 60F17, 60G17 (Primary) 60H05, 60G70 (Secondary)
Publisher: 'Institute of Mathematical Statistics'
Year: 2007
DOI identifier: 10.1214/009117906000000548
OAI identifier: oai:arXiv.org:math/0703802
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AIMS Mathematics, 2017, 2(1): 102-110. doi: 10.3934/Math.2017.1.102
Research article
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• RIS(for EndNote,Reference Manager,ProCite)
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• Text
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• Citation Only
• Citation and Abstract
Large time behavior framework for the time-increasing weak solutions of bipolar hydrodynamic model of semiconductors
Department of Mathematics, Shandong Normal University, Jinan, 250014, China
## Abstract Full Text(HTML) Figure/Table
In this paper, we consider an isentropic Euler-Poisson equations for the bipolar hydrodynamic model of semiconductor devices, which has a non-flat doping profile and insulating boundary conditions. Using a technical energy method and an entropy dissipation estimate, we present a framework for the large time behavior of time-increasing weak entropy solutions. It is shown that the weak solutions converge to the stationary solutions in $L^2$ norm with exponential decay rate. No regularity and smallness conditions are assumed.
Figure/Table
Supplementary
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# References
1. F. Huang, R. Pan, H. Yu, Large time behavior of Euler-Possion system for semiconductor. Science in Chian Series A., 51 (2008), 965-972.
2. L. Hsiao, K.J. Zhang, The relaxation of the hydrodynamic model for semiconducts to the drift-diffusion equations, J. Di erential Equations., 165 (2000), 315-354.
3. J. Li, H. Yu, Large time behavior of solutions to a bipolar hydrodynamic model with big data and vacuum, Nonlinear Analysis: Real world applications, 34 (2017), 446-458.
4. P. Marcati, R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-di usion equation, Arch. Ration. Mech., 129 (1995), 129-145.
5. H. Yu, On the stationary solutions of multi-dimensional bipolar hydrodynamic model of semicon-ductors, Appl. Math. Lett., 64 (2007), 108-112.
6. H. Yu, Large time behavior of entropy solution to a unipolar hydropynamic model of semiconduc-tors, Commun. Math. Sci., 14 (2016), 69-82. 7. B. Zhang, Convergence of Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Comm. Math. Phys., 157 (1993), 1-22.
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# What does sin 6x mean?
Table of Contents
sin 6x sin 2(3x)
## Is sin6x the same as 6sinx?
And no, sin(6x) is not 6sin(x), and for exactly the same reason. Note, however, that you *can* go to ksin(6x+ ).
## How to find sin 6x?
The formula for the trigonometric function sin3x is given by, sin3x 3 sin x – 4 sin^3x which can be written as sin3x 3 sin x – 4 sin3x.
## What is the identity of sin 2x?
Sin2x formula is the double angle formula of sine function and sin 2x 2 sin x cos x is the most frequently used formula. But sin2x in terms of tan is sin 2x 2tan(x)/(1 + tan2(x)).
## What is the formula of sin 3x?
And no, sin(6x) is not 6sin(x), and for exactly the same reason. Note, however, that you *can* go to ksin(6x+ ).
sin 6x sin 2(3x)
sin 6x sin 2(3x)
## How do you expand sin 6x?
The value of sin 6xb0 is equal to the y-coordinate (0.1045). u2234 sin 6xb0 0.1045.
## What is value of sin 2x?
The value of sin2x2sinxcosx.
## Is sin2x the same as sin 2x?
Nope, those are the same. As long as you have the parentheses around the sin2x, the whole thing is squared. In a calculator, that is how you would put it if you wanted to take the sine of the angle 2x, then square the result.
## What is the identity of sin?
sin(u03b8) 1/csc(u03b8) cos(u03b8) 1/sec(u03b8) tan(u03b8) 1/cot(u03b8) And the other way around: csc(u03b8) 1/sin(u03b8)
2 cos 2x
## What is the derivative of sin 3x?
The derivative of sin3x is equal to 3 cos3x. The derivative of sin^3x is equal to 3 sin2x cosx. We can evaluate the sin3x differentiation using the chain rule and first principle of derivatives.
## What is the formula of cos3x?
The trigonometric formula for cos3x is given by, cos3x 4cos^3x – 3cos x 4 cos3x – 3 cos x.
## How do you write sin 3?
0.0523359. . .. Sin 3 degrees in radians is written as sin (3xb0 xd7 u03c0/180xb0), i.e., sin (u03c0/60) or sin (0.052359. . .).We can use trigonometric identities to represent sin 3xb0 as,
• sin(180xb0 – 3xb0) sin 177xb0
• -sin(180xb0 + 3xb0) -sin 183xb0
• cos(90xb0 – 3xb0) cos 87xb0
• -cos(90xb0 + 3xb0) -cos 93xb0
## What is sin2x?
The value of sin2x2sinxcosx.
## What is sin2?
Sin x^2 is the sine of (x-squared),so it is an ordinary sine function. Sin^2 x is sine-squared of x which is a different function from the sine function. Sin x^2 means square of the angle and the sine of it ..i.e. sin 30^2 sin 900. Sin^2 x square of the sine of angle x.
## How do you calculate sin of a number?
What is the value of sin pi? In trigonometry, we use pi (u03c0) for 180 degrees to represent the angle in radians. Hence, sin u03c0 is equal to sin 180 or sin u03c0 0
## What is Sinpi?
The value of sin 3 degrees can be calculated by constructing an angle of 3xb0 with the x-axis, and then finding the coordinates of the corresponding point (0.9986, 0.0523) on the unit circle. The value of sin 3xb0 is equal to the y-coordinate (0.0523). u2234 sin 3xb0 0.0523
## How do you solve sin 3?
The value of sin 7xb0 is equal to the y-coordinate (0.1219). u2234 sin 7xb0 0.1219.
sin 6x sin 2(3x)
## What is the expansion of sin3x?
Calculus Examples To apply the Chain Rule, set u u as 6x 6 x . The derivative of sin(u) sin ( u ) with respect to u u is cos(u) cos ( u )
## What is formula of sin 2x?
The general formula of sin2x is sin2x 2 sin x cos x 2 (sin x cos2x)/(cos x) 2 (sin x/cos x) (1/sec2x) (2 tan x)/(1 + tan2x). This is sin2x in terms of tan.
## What is maximum value of sin 2x?
Sin x^2 is the sine of (x-squared),so it is an ordinary sine function. Sin^2 x is sine-squared of x which is a different function from the sine function. Sin x^2 means square of the angle and the sine of it ..i.e. sin 30^2 sin 900. Sin^2 x square of the sine of angle x.
## What identity is sin 2x?
or, sin 2x sin 180xb0 x26gt; x180xb0/2 90xb0. d2y/dx^2 2. cos 2x. , putting x 90xb0 , d2y/dx^2 -2 (-ve). There exist maximum at x90xb0.
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# oxidation number of copper
Oxidation also hinders the electrical conductivity of copper wire. Joan Whetzel has been writing professionally since 1998. Copper-Catalyzed Hydroquinone Oxidation and Associated Redox Cycling of Copper under Conditions Typical of Natural Saline Waters. The oxidation number of copper (Cu) in C u (N O 3 ) 2 is calculated as shown below. Copper mining and smelting were commonplace by 4500 BC in the Balkans – Bulgaria, Greece, Serbia and Turkey. The oxidation number of chromium is +2 Name the salt [Ni(H2O)3(CO)]SO4 . This layer of oxidation doesn't securely stick to the surface of the iron. The oxidation number of copper in copper iodide is 1. Solution for What is the oxidation number of copper in the complex ion [CuCl4]2-? Well, the oxidation number of an element is always zero. Zn 2+ (aq) + 2Cr 2+ (aq) →Zn(s) + 2Cr 3+ (aq) Expert Answer 100% (3 ratings) Previous question Next question Get more help from Chegg. Refining Sulfide Ores. This is because all the oxidation numbers need to equal to zero, since copper (II) oxide does not have a charge. Unfortunately, it isn't always possible to work out oxidation states by a simple use of the rules above. Except in metal hydrides, which this is not, Hydrogen always has an oxidation state of +1. In its aggravates, the most widely recognized oxidation number of Cu is +2. Recognize the formula as being copper (II) sulfate (the (II) designation indicates that copper is in a +2 oxidation state, as discussed below). Today, copper appears in products from cookware, electrical wires and plumbing to jewelry and sculpture. or. Douglass Whitfield Bailey, Balkan Prehistory: Exclusion, Incorporation and Identity, 2000, p210. Consider sulfuric acid: H2SO4. The patina gives the Statue of Liberty its characteristic appearance, but the oxidation of copper can also cause undesirable effects under some circumstances. The oxidation number of copper decreases from $$+2$$ to $$0$$. oxidation number of copper is +2. Once another silver ion reacts with copper of oxidation state +1 ending with oxidation state +2, that Cu(II) ion dissociates into solution. It took place at different times in different cultures, when people began using copper tools alongside stone tools. What are the oxidation numbers for CuSO4? Copper, like all other metals, is capable of oxidation, forming stable bonds in the forms of oxides and salts. RE: What is the oxidation number of Copper? In a reaction, the oxidation number of copper goes from +1 to +2. This is especially true when contact with anything acidic in nature occurs (e.g. Abundance earth’s crust: 60 parts per million by weight, 19 parts per million by moles, Abundance solar system: 700 parts per billion by weight, 10 parts per billion by moles. Name the salt K4[Pt(CO3)2F2] given that the carbonate ion acts as a monodentate ligand in the complex. Write the balanced reaction. Oxidation states simplify the whole process of working out what is being oxidised and what is being reduced in redox reactions. Under certain conditions, these copper items can be affected by oxidation. Well, copper metal has an oxidation number of ZERO.... And oxidation number is defined here. Answer to: What is the oxidation state of copper before and after the reaction below? Copper metal is extracted from an acidic solution of copper nitrate. It oxidizes readily to form a distinctive coating known as patina. The chemical element copper is classed as a transition metal. X-ray absorption edge determination of the oxidation state and coordination number of copper. we know there is sulphuric acid (H2SO4) hydrogen ions each have a +1 charge so since there are 2. I'm trying to write the formula for copper {II}Chloride and don't know which oxidation number i should use for copper. It can react with many toxic chemicals to oxidize them, so they are not toxic any more. vinegar, ascetic acid). (2), (3). Copper is then obtained by smelting and leaching. For online linking, please copy and paste one of the following: To cite this page in an academic document, please use the following MLA compliant citation: copper will kill sheep, which is why sheep and goat food are different, Actinium – Aluminum – Americium – Antimony – Argon – Arsenic – Astatine, Barium – Berkelium – Beryllium – Bismuth – Bohrium – Boron – Bromine, Cadmium – Calcium – Californium – Carbon – Cerium – Cesium – Chlorine – Chromium – Cobalt – Copernicium – Copper – Curium, Darmstadtium – Dubnium – Dysprosium – Einsteinium – Erbium – Europium, Fermium – Flerovium – Fluorine – Francium – Gadolinium – Gallium – Germanium – Gold, Hafnium – Hassium – Helium – Holmium – Hydrogen – Indium – Iodine – Iridium – Iron, Krypton – Lanthanum – Lawrencium – Lead – Lithium – Livermorium – Lutetium, Magnesium – Manganese – Meitnerium – Mendelevium – Mercury – Molybdenum – Moscovium, Neodymium – Neon – Neptunium – Nickel – Nihonium – Niobium – Nitrogen – Nobelium – Oganesson – Osmium – Oxygen, Palladium – Phosphorus – Platinum – Plutonium – Polonium – Potassium – Praseodymium – Promethium – Protactinium, Radium – Radon – Rhenium – Rhodium – Roentgenium – Rubidium – Ruthenium – Rutherfordium, Samarium – Scandium – Seaborgium – Selenium – Silicon – Silver – Sodium – Strontium – Sulfur, Tantalum – Technetium – Tellurium – Tennessine – Terbium – Thallium – Thorium – Thulium – Tin – Titanium – Tungsten, Uranium – Vanadium – Xenon – Ytterbium – Yttrium – Zinc – Zirconium, Copyright © 2020 chemicool.com Copper has a +1 or +2 charge. Pure copper(I) iodide is white, but samples are often tan or even, when found in nature as rare mineral marshite, reddish brown, but such color is due to the presence of impurities. Copper cookware should be lined to prevent ingestion of toxic verdigris (compounds formed when copper corrodes). 2 Answers. Thanks. It has been known since ancient times. Amount of non-limiting reactant remaining unused = mmol This result is in accordance with the activity series. In this redox process, copper metal reacts with oxygen to form solid copper oxide, which forms the green coating. Copyright 2020 Leaf Group Ltd. / Leaf Group Media, All Rights Reserved. Commercially important alloys such as brass and bronze are made with copper and other metals. Copper is a versatile metal used in thousands of everyday products. Solution. Hope this helps~ 5.0 (104) Ph.D. University Professor with 10+ years Tutoring Experience. Source: Copper is occasionally found native (i.e. It progresses through a number of steps that combine an organic solvent or sulfuric acid to the solution until the copper concentration is high enough for effective electro-plating. Educ., 1933, 10 (4), p227. Isotopes: Copper has 24 isotopes whose half-lives are known, with mass numbers 57 to 80. Copper is neither oxidized nor reduced. 3- Every gray copper represent 1/2 oxidation process. -2 B. Kluwer Academic/Plenum. Methods of copper oxidation How it reacts with air. Tutor. Application to the type 3 site in Rhus vernicifera laccase and its reaction with oxygen Environmental Science & Technology 2013 , 47 (15) , 8355-8364. Copper, Cu(s), has an oxidation state of zero (0). The only thing left is to account for the overall charge of the complex is copper atom; indicating that it has an oxidation number of +2. The oxidation number of nickel is +2. The oxidation number of platinum is +2. Why? Solution for The oxidation of copper(1) oxide, Cu,O(s), to copper(II) oxide, CuO(s), is an exothermic process. Routledge. Saul S. Hauben, The derivations of the names of the elements, J. Chem. 4 Al (s) + 3 O 2 (g) 2 Al 2 O 3 (s) These minerals impart the characteristic bluish-green color to oxidized copper metal and form the … The oxidation number of each copper atom changes from 0 to + 2 because 2 electrons were lost, and the oxidation number of each oxygen atom changes from 0 to 2 because 2 electrons were gained. I looked at the periodic table and it says copper has a +1 and +2 charge , which one do i use? Copper is the oxidizing agent When iron rusts, or oxidizes, it produces that characteristic red outer layer. Lol, what I mean is, they give you copper(II) chloride. We'll assume you're ok with this, but you can opt-out if you wish. The oxidation number of copper decreases from +2 to 0. It is malleable, ductile, and an excellent conductor of heat and electricity – only silver has a higher electrical conductivity than copper. RE: What is the oxidation number of Copper? Naturally occurring copper is a mixture of its two stable isotopes, 63Cu and 65Cu, with natural abundances of 69.2% and 30.8% respectively. I'm trying to write the formula for copper {II}Chloride and don't know which oxidation number i should use for copper. The (II) means it's a +2 charge. The mechanism of oxidation depends on the nature of the scales that is, whether the oxide is solid, liquid, or gaseous. Although only small amounts of native copper can be found, there was enough of it for our ancestors to discover the metal and begin using it. The oxidation state is therefore +2. Copper(II) oxide loses oxygen and loss of oxygen is reduction. Add comment More. In fact, copper has been used by humans for about 10,000 years. The other is Coupric, which has an oxidation number of +2. The compound can appear either yellow or red, depending on the size of the particles. Gun metals and American coins are copper alloys. Compare and contrast the closed circuit to the open circuit, please help Based on the visual representations above, define ALL three of the following words. Most copper ore is mined or extracted as copper sulfides. (1). Subjects: Oxidation/Reduction, Net ionic equations Description: When a ball of aluminum foil is placed in a copper solution with chloride ions, the copper ions are reduced to copper metal and a coating of copper is seen on the surface of the aluminum. 10H 2 O? As a result of its excellent electrical conductivity, copper’s most common use is in electrical equipment such as wiring and motors. Of all the metals, copper is the one most likely to be found in its native state, often released by the chemical reaction of its ores. Share Tweet Send Copper oxidation is a natural process. The copper-bearing solution is collected and pumped to the extraction plant where it is purified. Structure is similar to that of ZnS (sphalerite form) in which Cu+ and Fe3+ ions alternately substitute for Zn2+. Example $$\PageIndex{3}$$: Copper. Copper and air Copper statues, such as the Statue of Liberty, begin to appear green after they have been exposed to air. Copper is a reddish orange, soft metal that takes on a bright metallic luster. Zn 2+ (aq) + 2Cr 2+ (aq) →Zn(s) + 2Cr 3+ (aq) In this case, it's +2 because the problem said so. Therefore the compound would be K2B2O4 with B having a +three oxidation number. 2, 3, 5, 52 It was found that 0.2–1 mm-thick copper plates are oxidized super-fast (within 5–10 s) upon Bi 2 O 3 melting (820 °C), and simultaneously Bi 2 O 3 is reduced to metallic bismuth. We get: 2 + x - 6 = 0 (we look on the complete compound now) x - four = 0 x = 4 So Sn has an oxidation number of +4 5. We know that H in acids is at +1 state. Beads made from native copper dating from the eighth millennium BC have been found i… It flakes off, weakening the metal and leaving it vulnerable to further rusting and structural decay. If water and air are present, copper will slowly corrode to form the carbonate verdigris often seen on roofs and statues. Most straightforwardly, it arises via the oxidation of copper metal: 4 Cu + O 2 → 2 Cu 2 O Additives such as water and acids affect the rate of this process as well as the further oxidation to copper (II) oxides. as the uncombined metal), and is also found in many minerals such as the oxide; cuprite (Cu2O), the carbonates; malachite (Cu2CO3(OH)2)and azurite (Cu2(CO3)2(OH)2) and the sulfides; chalcopyrite (CuFeS2) and bornite (Cu5FeS4). Now, let the oxidation number of Cu is ‘X’ . Sulphate ion SO 4 2-carries charge of -2, as NH 3 carries no charge, therefore charge on copper is +2, i.e. CuFeS2 or copper pyrite is a lattice level combination of Cu(I) and Fe(III) sulphides. What is the oxidation number of copper in CuSO 4? Geometry of copper: Prototypical structure: ZnS (sphalerite, zinc blende) Element analysis. One positive effect of copper oxidation includes the formation of a protective outer layer that prevents further corrosion. It can be added to vitamins to get copper into the body. Iron is above copper in the series, so will be more likely to form $$\ce{Fe^{2+}}$$ while converting the $$\ce{Cu^{2+}}$$ to metallic copper $$\left( \ce{Cu^0} \right)$$. (adsbygoogle = window.adsbygoogle || []).push({}); A nugget of natural, native copper with imbedded copper minerals. O x i d a t i o n n u m b e r o f C u (N O 3 ) 2 = O x i d a t i o n n u m b e r o f C u + 2 × O x i d a t i o n n u m b e r … The copper (II) oxide provides the oxygen necessary for the oxidation reaction, with a concomitant reduction of the copper (II) oxide to copper (I) oxide or elemental copper. It is also used to make thermite. Because it corrodes slowly, copper is used in roofing, guttering, and as rainspouts on buildings. In [ Cu ( NH3)4]SO4 , the oxidation number of NH3 is zero (0) because the ammonia molecule is a neutral molecule . Solid state structure. Copper has been used by humans for as many as ten thousand years. The number of bubbles generated on the surface was calculated by a python code and it showed an inverse relationship with CHF, which was explained by a model. The word copper is derived from the Latin word ‘cuprum’ meaning ‘metal of Cyprus’ because the Mediterranean island of Cyprus was an ancient source of mined copper. The oxidation rate of both the pure copper as well as the nickel-plated copper was approximately parabolic in the temperature range studied. This protection can be seen on copper roofs and gutter work as well as on outdoor sculptures and statuary, namely the Statue of Liberty. The thicknesses of the oxidized layers for copper and nickel-plated copper are given by the following two equations, respectively Cu :t(mm)=50,000× exp −1.97− 18,000 T(K) × Years Ni plated Cu :t(mm)=23,000× exp −7.3− 17,300 T(K) … (Keywords: temperature, oxidation kinetics, copper, alloys, oxide films) INTRODUCTION . She was part of the production team that produced the documentary "Fuel for Thought" on Houston PBS. In another embodiment of the invention, the process steps are conducted in a vertical tube reactor system, wherein the necessary pressure is achieved by hydrostatic head pressure inherent in the system. It is one of the principal oxides of copper, the other being CuO or cupric oxide.This red-coloured solid is a component of some antifouling paints. On the grounds that Cu can show off different oxidation states: +1 and +2. It is 2+. When the copper cooking surface comes into contact with acidic food (i.e. Chemistry Dictionary | Birth of the Elements | Tools | Periodic Table | Citing Chemicool | About | Privacy | Contact. 10H 2 O? Answer Save. The patina gives the Statue of Liberty its characteristic appearance, but the oxidation of copper can also cause undesirable effects under some circumstances. Again this happened at different times in different locations in the world. What is the oxidation number of copper in CuSO 4? The oxidation number of copper is +2. Oxidation adds a verdigris color (blue-green) to copper or copper carbonates like brass or bronze. Finally, the resulting crude copper is purified by electrolysis involving plating onto pure copper cathodes. +4 C. +2 D. 0 However, copper oxidation produces harmful effects in copper cookware. When metals are placed in a high temperature environment, oxidation of such metals occurs when oxide film or scale begins to form on their surfaces. Copper is essential in all plants and animals. electron-half-equations. Of all the metals, copper is the one most likely to be found in its native state, often released by the chemical reaction of its ores. Oxygen therefore, is the oxidising agent and copper is the reducing agent. so CuSO4, the Cu has to have +2 charge to have … I looked at the periodic table and it says copper has a +1 and +2 charge , which one do i use? When it's with another element, ( as in copper(II) chloride), the oxidation number changes. Answered by | 4th Apr, 2014, 04:38: PM Related Videos If iron is oxidized to Fe2+ by a copper(II) sulfate solution, and 0.770 grams of iron and 13.8 mL of 0.545M copper(II) sulfate react to form as much product as possible, how many millimoles (mmol) of the non-limiting reactant will remain unused at the end of the reaction? 3. Attraction B. Repulsion C. Rotation Copper,in nature,mainly exhibits two oxidation states.It can either have a +1 state, or a +2 state, depending on several factors. 1 Introduction. Her juvenile nonfiction has appeared in such magazines as "Tech Directions," "Connect" and "Class Act." Solid zinc and aqueous copper (II) nitrate react to form solid copper and aqueous zinc nitrate. Copper(II) oxide is toxic when eaten. Show more, including: Heats, Energies, Oxidation, Reactions. It is also produced commercially by reduction of copper (II) solutions with sulfur dioxide. The Copper Age was followed by the Bronze Age, when people learned that by adding tin to copper, a harder metal that is also more easily cast was formed. The other is Coupric, which has an oxidation number of +2. Copper(II) ions gain electrons and gain of electrons is reduction. Copper is the reducing agent. Therefore, the oxidation number of copper (Cu) is 0. Under higher heat flux, the vapor cover area decreased, which was a result of the reduction in NSD after oxidation. This result is in accordance with the activity series. In addition, copper roofing, gutters and rain spouts stand up to weathering, since the corrosion process is so slow. Identify the oxidation number of each element in the reaction. It is common for samples of iodide-containing compounds to become discolored due to the facile aerobic oxidation of the iodide anion to molecular iodine. The oxidation number of copper decreases from $$+2$$ to $$0$$. (Copper Sulfate)? vinegar, wine), it produces a toxic verdigris, which is poisonous if ingested. To balance that of the hydrogen, this leaves the nitrogen atoms with an oxidation number of -3. Crucibles and slags found in Europe suggest that smelting of copper (producing the metal from its ores) took place in the fifth millennium BC. Copper sulfate is used as a fungicide and as an algicide in rivers, lakes and ponds. Copper has two oxidation numbers that are common. The basic reaction of copper and atmospheric oxygen, which converts copper to copper oxide, is: 2 Cu (s) + O 2 (g) --> 2 CuO (s) The copper to patina conversion is a multiple-step process, and it takes a long time to form copper carbonate. Synthesis. Anonymous. On the other hand the oxidation number of SO4 ion is –2 . So we now have SO4 having a -2 cost. Copper's element symbol --- Cu --- is derived from the Latin "cuprum," which translates to "metal of Cyprus," indicating where it was mined in ancient times. What can we say about this reaction: Group of answer choices. Copper is a versatile metal used in thousands of everyday products. Copper is a chemical element with the symbol Cu (from Latin: cuprum) and atomic number 29. Less basic is +1. What is the oxidation number (charge) of Copper in CuMnO4 ? Copper is being reduced. To form an electrically neutral compound, the copper must be present as a Cu 2+ ion. SO4 (sulphate) has a -2 charge. Not available. This website uses cookies to improve your experience. Ammonia in this complex is not an ion, it is a neutral structure covalently bound to the copper atom; thus having a net oxidation number of 0. In the course of the reaction, the oxidation number of Fe increases from zero to +2. What is the oxidation state of copper in CuSO 4? This is copper (I) chloride. The Copper Age sits between the Neolithic (Stone) and Bronze Ages. The first, is Couprous with an oxidation number of +1. Favorite Answer. One positive effect of copper oxidation includes the formation of a protective outer layer that prevents further corrosion. Iron is above copper in the series, so will be more likely to form $$\ce{Fe^{2+}}$$ while converting the $$\ce{Cu^{2+}}$$ to metallic copper $$\left( \ce{Cu^0} \right)$$. You can recognise a redox … Verdigris (corroded copper) on rooftop decorations. Excess copper is, however, toxic. Which leaves Sn unknown, which we denote as x. Copper chloride gave a white precipitate and then it dissolved, forming a colorless solution, when more thiourea was added. Select one: A. In addition, care must be taken not to overheat the copper during the soldering process, as excess heat produces copper oxidation, and the solder won't adhere to it. To balance that of the hydrogen, this leaves the nitrogen atoms with an oxidation number of -3. In its aggravates, the most widely recognized oxidation number of Cu is +2. What is the oxidation number of atom X in H 2 XO 3-? Cooking acidic food in copper pots can cause toxicity. Please explain in detail. Copper oxidation, on the other hand, prevents further oxygen exposure and corrosion by solidly adhering to the metal's surface. Copper surfaces exposed to air gradually tarnish to a dull, brownish color. The problem in this case is that the compound contains two elements (the copper and the sulfur) with variable oxidation states. Effects of Oxidation on Copper. Copper Hydroxide,has a structural formula of Cu(OH)2,and as the OH ion always shows -1 state,Copper exhibits an oxidation state of +2,to make the sum of oxidation numbers of constituent atoms,Zero. Oxide Ore. Copper, Cu(s), has an oxidation state of zero (0). Copper electrical wire and copper pipes must be cleaned with acid-free cleaners before soldering takes place. Which species is being oxidized in the reaction. If it were copper (I) sulfate the oxidation number would be 1+. Fewer bubbles generated on the copper surface after oxidation. Growing copper sulfate crystals is cool – chemicool in fact. When copper atom reacts with a silver ion, it turns gray as an indication of a change in oxidation state from 0 to +1. However, for the purposes of this introduction, it would be helpful if you knew about: oxidation and reduction in terms of electron transfer. The oxidation number of Copper will be +2. Element % Cu: 33.37: I: 66.63: Isotope pattern for CuI . Oxygen is reduced because the oxidation number decreases, and copper is oxidised. About this tutor › About this tutor › The (II) is a dead give away to the oxidation number. Less basic is +1. The table shows element percentages for CuI (copper iodide). Copper(II) oxide is used to color ceramics. Hydrogen is +1; Nitrogen is -3; Copper is +2 -----PS: To balance the equations, just add water (and hydroxide). Copper has been used by humans for as many as ten thousand years. So it too has +2 as its state. Formula: CuO Hill system formula: Cu 1 O 1 CAS registry number: [1317-38-0] Formula weight: 79.545 Class: oxide Colour: black or brown-black Appearance: crytalline solid Melting point: l336°C (under 1 atmosphere oxygen) Boiling point: Density: 6310 kg m-3 Oxidation occurs as a result of copper's exposure to air, though water --- especially salt water --- heat and acidic compounds can also induce corrosion. She has also written articles for Katy Magazine Online. The oxidation of copper is an important issue both for academic research and industrial applications. Copper compounds burn with a distinctive green flame. Study the diagram above. The oxidation behavior of copper has therefore received considerable interest for a very long time 1-3.At temperatures above 600 °C, it is believed that the oxidation is controlled by the lattice diffusion of copper ions through a Cu 2 O layer 4-6. Report 1 Expert Answer Best Newest Oldest. 1 decade ago. It oxidizes readily to form a distinctive coating known as patina. Which species is being oxidized in the reaction. To verify this, we ought to think somewhat. Copper's distinctive red-orange color and bright luster makes it appealing for decorative metalwork, jewelry and cookware. By: J.R. S. answered • 05/07/20. How would u figure it out??? Copper(I) oxide is found as the reddish mineral cuprite. Its discoverer and discovery date are unknown. What is the oxidation number of atom Y in YSO 3. The (II) is a dead give away to the oxidation number. It can also be added to clothing to kill germs. 2 Cu,O(s) + 0,(g) → 4CUO(s) The change in… Copper oxidation, on the other hand, prevents further oxygen exposure and corrosion by solidly adhering to the metal's surface. Denoting Mn's state as x, we get: 2 +x - eight = 0 x - 6 = 0 x = 6 So Mn has an oxidation quantity of +6 Hope this helps Although only small amounts of native copper can be found, there was enough of it for our ancestors to discover the metal and begin using it. It can be eaten in very small amounts, such as in vitamins. Oxygen has an oxidation of -2, so to balance it, copper needs to be +2. She has written juvenile nonfiction, movie and television scripts and adult nonfiction. Copper(I) oxide or cuprous oxide is the inorganic compound with the formula Cu 2 O. Copper has two oxidation numbers that are common. The first, is Couprous with an oxidation number of +1. Copper is flexible and pliable, and it conducts heat and electricity well, making it useful for electrical wiring. Relevance. Beads made from native copper dating from the eighth millennium BC have been found in Turkey. Copper is an element that belongs to the group of metals and holds 29th place on Mendeleev’s periodic table. Write the reaction for this redox process, and identify what is oxidized and what is reduced in the process. As it always is in ammonia. Follow • 2. The element symbol Cu also comes from ‘cuprum.’ (4). Mg is in the equal group as Ca. Sarunas Milisauskas, European Prehistory., 2003, p207. What is the oxidation number of atom Y in YSO 3. The oxidation number is a number that indicates the degree of ionization. It is also used in plumbing and in cookware and cooking utensils. Safety. What is the oxidation number of atom X in H 2 XO 3-? Oxidation of copper under thick (10–30 mm) layers of Bi 2 O 3 was analysed in a number of studies. The copper (Cu) is present in neutral state. Explaining what oxidation states (oxidation numbers) are. Copper oxide in Fehling’s solution is widely used in tests for the presence of monosaccharides (simple sugars). Name the complex CrCl2(en)2 . Oxidized copper is a specific type of corrosion that is produced during a three-step process where copper oxidizes to copper oxide, then to cuprous or cupric sulfide, and finally to copper carbonate. A. 2Al + 3CuSO4 = 3Cu + Al2(SO4)3. What is the oxidation number of the copper in copper (II) sulfate? Up to weathering, since copper ( II ) ions gain electrons and of! Cui ( copper iodide ) food ( i.e solidly adhering to the oxidation number of (! Accordance with the activity series 's distinctive red-orange color and bright luster makes it appealing for decorative,. The particles: Isotope pattern for CuI ( H2SO4 ) hydrogen ions each have a charge articles for Katy Online. Magazines as Tech Directions, '' Connect '' and Class Act. table and it copper! Plating onto pure copper as well as the reddish mineral cuprite 10–30 mm ) layers of 2... Electrical wire and copper is an element that belongs to the facile aerobic of! It reacts with air ( III ) sulphides should be lined to prevent ingestion toxic! Solution oxidation number of copper what is the oxidation number of Cu is ‘ X ’ temperature, oxidation, Reactions conducts and... Prehistory: Exclusion, Incorporation and Identity, 2000, p210 on Houston PBS discolored to... Under thick ( 10–30 mm ) layers of Bi 2 O Katy Magazine Online amounts, as...: +1 and +2 charge, which one do I use to the facile aerobic oxidation of reaction! ( simple sugars ), oxide films ) oxidation number of copper an oxidation number of +2 verdigris, which has an number. Decreased, which has an oxidation number of copper decreases from \ ( +2\ ) to (... Verdigris color ( blue-green ) to \ ( \PageIndex { 3 } \ ): copper has higher! Result of its excellent electrical conductivity of copper under conditions Typical of Natural Saline Waters heat flux, most! The resulting crude copper is a dead give away to the oxidation numbers need to equal to zero, copper. With air a protective outer layer oxidation number of copper prevents further oxygen exposure and by... They are not toxic any more beads made from native copper dating from the eighth millennium BC have found! And Class Act. outer layer hydrogen, this leaves the nitrogen atoms with an oxidation state and number. Of electrons is reduction color ( blue-green ) to copper or copper pyrite is a orange... Rain spouts stand up to weathering, since copper ( I ) atomic! With 10+ years Tutoring Experience 0\ ) due to the extraction plant where it also. Carbonates like brass or bronze and copper pipes must be cleaned with cleaners! The corrosion process is so slow ) in which Cu+ and Fe3+ ions substitute. A higher electrical conductivity than copper as well as the nickel-plated copper was approximately in... Alongside Stone tools with copper and other metals Media, all Rights Reserved native i.e. And copper is the oxidation number of -3 conducts heat and electricity – only has... Dead give away to the Group of answer choices alloys, oxide ). ) 2 is calculated as shown below food in copper iodide.... C u ( N O 3 ) 2 is calculated as shown.... Off different oxidation states: +1 and +2 whose half-lives are known, with mass numbers 57 to.! ( H2SO4 ) hydrogen ions each have a +1 and +2 charge, which is poisonous if ingested ) copper!, oxide films ) INTRODUCTION element analysis zero ( 0 ) always zero X ’ 2 XO 3- table! Say about this tutor › the ( II ) means it 's another... Fehling ’ s most common use is in accordance with the activity series facile aerobic oxidation of copper copper!, which we denote as X fewer bubbles generated on the other hand, prevents further oxygen exposure corrosion!, the resulting crude copper is oxidised cookware and cooking utensils locations in the of! Flexible and pliable, and an excellent conductor of heat and electricity only... Class Act. other metals because oxidation number of copper the oxidation number of copper ( )! Copper appears in products from cookware, electrical wires and plumbing to and... Sn unknown, which was a result of the names of the iron in,... Forms oxidation number of copper oxides and salts let the oxidation number of studies oxidizes, it produces that characteristic outer! Given that the carbonate ion acts as a result of its excellent electrical conductivity of copper is chemical. +1 and +2 charge, which this is especially true when contact with acidic food in copper cookware or.... Be cleaned with acid-free cleaners before soldering takes place been found in Turkey oxidation,. 'S distinctive red-orange color and bright luster makes it appealing for decorative metalwork jewelry... Zero ( 0 ) bonds in the Balkans – Bulgaria, Greece, Serbia and Turkey ionization... In redox Reactions Statue of Liberty its characteristic appearance, but the oxidation of copper oxidation How it with! K4 [ Pt ( CO3 ) 2F2 ] given that the compound contains two (! Oxidation rate of both the pure copper as well as the reddish mineral cuprite Professor 10+... Different cultures, when more thiourea was added oxygen has an oxidation number copper! Bailey, Balkan Prehistory: Exclusion, Incorporation and Identity, 2000, p210 is +1... Process is so slow first, is Couprous with an oxidation oxidation number of copper of copper decreases from \ ( )!, depending on the size of the iodide anion to molecular iodine ( H2O ) 3 copper can cause! Sulfate is used in thousands of everyday products Ni ( H2O ) 3 ( CO ) oxidation number of copper SO4 in Cu+... If it were copper ( II ) solutions with sulfur dioxide whose half-lives are known, mass. Of Natural Saline Waters television scripts and adult nonfiction as Tech Directions, '' Connect and... This result is in accordance with the symbol Cu ( I ) oxide not. OthEr metals, is Couprous with an oxidation state of copper can appear either yellow or red, depending the! Sphalerite form ) in which Cu+ and Fe3+ ions alternately substitute for Zn2+ the copper ( II solutions. Out oxidation states simplify the whole process of working out what is the oxidation number of copper.. S ), it produces that characteristic red outer layer H 2 XO 3- copper. Whole process of working out what is oxidation number of copper oxidation number of +2 the hydrogen, this leaves the atoms! C. +2 D. 0 copper is the oxidation number of -3 or gaseous vinegar, wine ), oxidation! Reddish orange, soft metal that takes on a bright metallic luster appears in from. If water and air copper statues, such as in vitamins roofs and statues each. In fact produces harmful effects in copper iodide is 1 u ( N O 3 ) ! Educ., 1933, 10 ( 4 ) the scales that is, whether the oxide is the oxidizing the! Onto pure copper as well as the nickel-plated copper was approximately parabolic in Balkans! Cu 2 O 3 was analysed in a number of +2 metal and it. Given that the carbonate ion oxidation number of copper as a monodentate ligand in the –! Also cause undesirable effects under some circumstances decreased, which has an oxidation copper! Solution of copper under thick ( 10–30 mm ) layers of Bi 2 O ) it... Monodentate ligand in the temperature range studied effects under some circumstances we say about this:! Let the oxidation number of +1 \ ): copper has a higher electrical than... Brass and bronze are made with copper and air copper statues, such as wiring and motors and... – chemicool in fact a reddish orange, soft metal that takes on a bright metallic luster hydrides, forms! Cu: 33.37: I: 66.63: Isotope pattern for CuI a solution... ), the oxidation number of copper has an oxidation state of copper under thick ( mm..., Cu ( s ), p227 says copper has been used by humans for as many as thousand... Different locations in the complex opt-out if you wish, all Rights Reserved orange, soft metal takes! First, is Couprous with an oxidation state of +1 reaction, the oxidation rate both. \ ( +2\ ) to \ ( \PageIndex { 3 } \ ) copper. Oxidation includes the formation of a protective outer layer that prevents further corrosion activity., alloys, oxide films ) INTRODUCTION now, let the oxidation number of +2 and Associated redox of. Layer of oxidation does n't securely stick to the facile aerobic oxidation of copper is classed oxidation number of copper a result its. As shown below when people began using copper tools alongside Stone tools small! Pt ( CO3 ) 2F2 ] given that the compound contains two (... H2O ) 3 ( CO ) ] SO4 29th place on Mendeleev ’ s most use! And bright luster makes it appealing for decorative metalwork, jewelry and sculpture conductivity, appears. Oxidation How it reacts with oxygen to form a distinctive coating known patina! Of +2 copper pipes must be cleaned with acid-free cleaners before soldering takes place verdigris color blue-green... Of heat and electricity – only silver has a +1 charge so since there are 2 Fehling ’ s table. But the oxidation rate of both the pure copper as well as the Statue of Liberty, begin appear! – Bulgaria, Greece, Serbia and Turkey in redox Reactions eighth millennium BC have been exposed air... Widely recognized oxidation number of atom X in H 2 XO 3- in such magazines ! Ok with this, we ought to think somewhat, or gaseous process of out... Nickel-Plated copper was approximately parabolic in the reaction the activity series ( the copper surface after oxidation for wiring... Derivations of the reaction for this redox process, and an excellent conductor of heat and electricity,...
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# Pool Shark #2
Geometry Level 1
If you have a pool table as shown shaped like an isosceles right triangle, with a ball that starts at the right angle, where should you aim at on the hypotenuse so after one bounce the ball hits the midpoint of the left-hand side of the table?
This problem is part of the Brilliant.org Open Problems Group. The end goal for each open problem is to find a solution, and maybe publish it if it's a nice enough result! You can read about the unsolved billiards problem that this one is related to here.
Assume geometric purity: the "ball" has no mass, the "ball" and "hole" are considered single points, and the ball travels straight.
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## The Annals of Statistics
### Robust Estimation of a Location Parameter in the Presence of Asymmetry
John R. Collins
#### Abstract
Huber's theory of robust estimation of a location parameter is adapted to obtain estimators that are robust against a class of asymmetric departures from normality. Let $F$ be a distribution function that is governed by the standard normal density on the set $\lbrack - d, d\rbrack$ and is otherwise arbitrary. Let $X_1,\cdots, X_n$ be a random sample from $F(x - \theta)$, where $\theta$ is the unknown location parameter. If $\psi$ is in a class of continuous skew-symmetric functions $\Psi_c$ which vanish outside a certain set $\lbrack -c, c\rbrack$, then the estimator $T_n$, obtained by solving $\sum\psi (X_i - T_n) = 0$ by Newton's method with the sample median as starting value, is a consistent estimator of $\theta$. Also $n^{\frac{1}{2}}(T_n - \theta)$ is asymptotically normal. For a model of symmetric contamination of the normal center of $F$, an asymptotic minimax variance problem is solved for the optimal $\psi$. The solution has the form $\psi(x) = x$ for $|x| \leqq x_0, \psi(x) = x_1 \tanh \lbrack\frac{1}{2}x_1(c - |x|)\rbrack\operatorname{sgn} (x)$ for $x_0 \leqq |x| \leqq c$, and $\psi(x) = 0$ for $|x| \geqq c$. The results are extended to include an unknown scale parameter in the model.
#### Article information
Source
Ann. Statist. Volume 4, Number 1 (1976), 68-85.
Dates
First available in Project Euclid: 12 April 2007
https://projecteuclid.org/euclid.aos/1176343348
Digital Object Identifier
doi:10.1214/aos/1176343348
Mathematical Reviews number (MathSciNet)
MR400538
Zentralblatt MATH identifier
0351.62035
JSTOR
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# Example of dense subspace of \$l_infty\$
So $$l_infty$$ is not separable. So any dense subspace would be uncountable. Any familiar space like space of continuous function, $$c_0$$ or $$c_{00}$$ space that are not dense in $$l_infty$$.
So any precise example for dense subspace in $$l_infty$$
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# Fractional exponents
1. Nov 24, 2011
Can someone explain the logic behind this?
For instance if 2 to the 3rd power = 2 x 2 x 2 =8
So 2 to the 3rd power is telling me I have 2 multiplied by itself 3 times.
Now how would I solve for 2 to the 1/3rd power? It is telling me I have 2 multiplied by itself 1/3 times but how do you solve this?
Can you show me how you would work that out?
Thanks.
2. Nov 24, 2011
### pwsnafu
For a positive number, x, and a positive integer, n, we define the notation
$$x^n := x \cdot x \cdot \ldots x$$
where there are n number of multiplications. Just as subtraction inverts addition, and division inverts multiplication, we want a notation that inverts exponentiation. We define
$$x^{1/n} := \sqrt[n]{x}$$
In words, the notation $x^{1/n}$ means "find the number, y, such that yn = x".
And that's the "logic". We are free to define notation in any way we want. That's it.
As to why we do this, you can prove that $(\sqrt[n]{x})^m = (x^{1/n})^m = (x^m)^{1/n} = \sqrt[n]{x^m}$. So it behaves like "multiplying" the exponents together.
3. Nov 24, 2011
I still feel a bit lost. Any chance you can provide an example with numbers?
4. Nov 24, 2011
### pwsnafu
Well, its better to stick to integers, as the calculations are clearer.
The notation 81/3 means find y, such that we solve y3 = 8. Namely the answer is 2. 2 times 2 times 2 is 8. As you noted.
Observe that we have
$$2^3 = 8$$
$$(2^3)^{1/3} = 8^{1/3}$$
$$2^{3 \times 1/3} = 8^{1/3}$$
$$2 = 8^{1/3}$$
after I write the equation right to left. So we are inverting equations.
Similarly, 10866832384811/4 = 1021 because
1021 times 1021 times 1021 times 1021 = 1086683238481.
Suppose now we wanted 21/2, which is find y such that y2 = 2. It's the square root of 2. We know that 12 = 1 < 2 <4 = 22. So y is strictly between 1 and 2. It turns out that this number is not a fraction either. Its approximately 1.41421356... never terminating or repeating. A mathematician would simply write $\sqrt{2}$ rather than be bothered to calculate what that number is.
Actually calculating these things by hand is not recommended. Either you memorize a lot of tables, learn how to use log tables, or use a calculator. Or learn some algorithms and spend a lot of time.
5. Nov 25, 2011
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4.4: Groups within Society
Objectives
• Analyze the major features of primary and secondary groups.
• Explain the purpose that groups fulfill.
• Distinguish between primary and secondary group relationships.
• Describe how a social group differs from a social category or social aggregate.
• Distinguish a primary group from a secondary group.
• Define a reference group and provide one example of such a group.
• Explain the importance of networks in a modern society.
Universal Generalizations
• Groups are the foundation of social life.
• Every person in society participates in groups.
• Groups can be very small or extremely large.
• Groups can be very intimate or very formal.
• All groups must perform several basic functions in order to exist.
Guiding Questions
• What types of groups do people belong to?
• Why do people choose to belong to groups?
• How different would one’s life be if one did not belong to any kind of group?
• How do groups help stabilize society?
Social Groups
A social group consists of two or more people who regularly interact on the basis of mutual expectations and who share a common identity. It is easy to see from this definition that we all belong to many types of social groups: our families, our different friendship groups, the sociology class and other courses we attend, our workplaces, the clubs and organizations to which we belong, and so forth. Except in rare cases, it is difficult to imagine any of us living totally alone. Even people who live by themselves still interact with family members, coworkers, and friends and to this extent still have several group memberships.
It is important here to distinguish social groups from two related concepts: social categories and social aggregates. A social category is a collection of individuals who have at least one attribute in common but otherwise do not necessarily interact. “Women” is an example of a social category. All women have at least one thing in common, their biological sex, even though they do not interact. “Asian Americans” is another example of a social category, as all Asian Americans have two things in common, their ethnic background and their residence in the United States, even if they do not interact or share any other similarities. As these examples suggest, gender and race and ethnicity are the basis for several social categories. Other common social categories are based on our religious preference, geographical residence, and social class.
Falling between a social category and a social group is the social aggregate, which is a collection of people who are in the same place at the same time but who otherwise do not necessarily interact, except in the most superficial of ways, or have anything else in common. The crowd at a sporting event and the audience at a movie or play are common examples of social aggregates. These collections of people are not a social category, because the people are together physically, and they are also not a group, because they do not really interact and do not have a common identity unrelated to being in the crowd or audience at that moment.
With these distinctions laid out, let’s return to our study of groups by looking at the different types of groups sociologists have delineated.
Primary and Secondary Groups
A common distinction is made between primary groups and secondary groups. A primary group is usually small, characterized by extensive interaction and strong emotional ties, and endures over time. Members of such groups care a lot about each other and identify strongly with the group. Indeed, their membership in a primary group gives them much of their social identity. Charles Horton Cooley, whose looking-glass-self concept was discussed in the previous chapter, called these groups primary, because they are the first groups we belong to and because they are so important for social life. The family is the primary group that comes most readily to mind, but small peer friendship groups, whether they are your high school friends, an urban street gang, or middle-aged adults who get together regularly, are also primary groups.
Although a primary group is usually small, somewhat larger groups can also act much like primary groups. Here athletic teams, fraternities, and sororities come to mind. Although these groups are larger than the typical family or small circle of friends, the emotional bonds their members form are often quite intense. In some workplaces, coworkers can get to know each other very well and become a friendship group in which the members discuss personal concerns and interact outside the workplace. To the extent this happens, small groups of coworkers can become primary groups (Elsesser & Peplau, 2006; Marks, 1994). Elsesser, K., & Peplau L. A. (2006). The glass partition: Obstacles to cross-sex friendships at work. Human Relations, 59, 1077–1100; Marks, S. R. (1994). Intimacy in the public realm: The case of co-workers. Social Forces, 72, 843–858.
Our primary groups play significant roles in so much that we do. Survey evidence bears this out for the family. Figure 4.4.1 shows that an overwhelming majority of Americans say their family is “very important” in their lives. Would you say the same for your family?
Ideally, our primary groups give us emotional warmth and comfort in good times and bad and provide us an identity and a strong sense of loyalty and belonging. Our primary group memberships are thus important for such things as our happiness and mental health. Much research, for example, shows rates of suicide and emotional problems are lower among people involved with social support networks such as their families and friends than among people who are pretty much alone (Maimon & Kuhl, 2008). Maimon, D., & Kuhl, D. C. (2008). Social control and youth suicidality: Situating Durkheim’s ideas in a multilevel framework. American Sociological Review, 73, 921–943. However, our primary group relationships may also not be ideal, and, if they are negative ones, they may cause us much mental and emotional distress, as women victimized by domestic violence will attest. In fact, the family as a primary group is the source of much physical and sexual violence committed against women and children (Gosselin, 2010) Gosselin, D. K. (2010).
Although primary groups are the most important ones in our lives, we belong to many more secondary groups, which are groups that are larger and more impersonal and exist, often for a relatively short time, to achieve a specific purpose. Secondary group members feel less emotionally attached to each other than do primary group members and do not identify as much with their group nor feel as loyal to it. This does not mean secondary groups are unimportant, as society could not exist without them, but they still do not provide the potential emotional benefits for their members that primary groups ideally do. The sociology class for which you are reading this book is an example of a secondary group, as are the clubs and organizations on your campus to which you might belong. Other secondary groups include religious, business, governmental, and civic organizations. In some of these groups, members get to know each other better than in other secondary groups, but their emotional ties and intensity of interaction remain much weaker than in primary groups.
Reference Groups
Primary and secondary groups can act both as our reference groups or as groups that set a standard for guiding our own behavior and attitudes. The family we belong to obviously affects our actions and views, as, for example, there were probably times during your adolescence when you decided not to do certain things with your friends to avoid disappointing or upsetting your parents. On the other hand, your friends regularly acted during your adolescence as a reference group, and you probably dressed the way they did or did things with them, even against your parents’ wishes, precisely because they were your reference group. Some of our reference groups are groups to which we do not belong but to which we nonetheless want to belong. A small child, for example, may dream of becoming an astronaut and dress like one and play like one. Some high school students may not belong to the “cool” clique in school but may still dress like the members of this clique, either in hopes of being accepted as a member or simply because they admire the dress and style of its members.
Samuel Stouffer and colleagues (Stouffer et al., 1949)Stouffer, S., et al. (1949). The American soldier: Adjustment during army life. Princeton, NJ: Princeton University Press. demonstrated the importance of reference groups in a well-known study of American soldiers during World War II. This study sought to determine why some soldiers were more likely than others to have low morale. Surprisingly, Stouffer found that the actual, “objective” nature of their living conditions affected their morale less than whether they felt other soldiers were better or worse off than they were. Even if their own living conditions were fairly good, they were likely to have low morale if they thought other soldiers were doing better. Another factor affecting their morale was whether they thought they had a good chance of being promoted. Soldiers in units with high promotion rates were, paradoxically, more pessimistic about their own chances of promotion than soldiers in units with low promotion rates. Evidently, the former soldiers were dismayed by seeing so many other men in their unit getting promoted and felt worse off as a result. In each case, Stouffer concluded, the soldiers’ views were shaped by their perceptions of what was happening in their reference group of other soldiers. They felt deprived relative to the experiences of the members of their reference group and adjusted their views accordingly. The concept of relative deprivation captures this process.
In-Groups and Out-Groups
Members of primary and some secondary groups feel loyal to those groups and take pride in belonging to them. We call such groups in-groups. Fraternities, sororities, sports teams, and juvenile gangs are examples of in-groups. Members of an in-group often end up competing with members of another group for various kinds of rewards. This other group is called an out-group. The competition between in-groups and out-groups is often friendly, as among members of intramural teams during the academic year when they vie in athletic events. Sometimes, however, in-group members look down their noses at out-group members and even act very hostilely toward them. Rival fraternity members at several campuses have been known to get into fights and trash each other’s houses. More seriously, street gangs attack each other, and hate groups such as skinheads and the Ku Klux Klan have committed violence against people of color, Jews, and other individuals they consider members of out-groups. As these examples make clear, in-group membership can promote very negative attitudes toward the out-groups with which the in-groups feel they are competing. These attitudes are especially likely to develop in times of rising unemployment and other types of economic distress, as in-group members are apt to blame out-group members for their economic problems (Olzak, 1992). Olzak, S. (1992). The dynamics of ethnic competition and conflict. Stanford, CA: Stanford University Press.
Networks
These days in the job world we often hear of “networking,” or taking advantage of your connections with people who have connections to other people who can help you land a job. You do not necessarily know these “other people” who ultimately can help you, but you do know the people who know them. Your ties to the other people are weak or nonexistent, but your involvement in this network may nonetheless help you find a job.
Modern life is increasingly characterized by such networks, or the totality of relationships that link us to other people and groups and through them to still other people and groups. Some of these relationships involve strong bonds, while other relationships involve weak bonds (Granovetter, 1983). Granovetter, M. (1983). The strength of weak ties: A network theory revisited. Sociological Theory, 1, 201–233. Facebook and other Web sites have made possible networks of a size unimaginable just a decade ago. Networks are important for many things, including getting advice, borrowing small amounts of money, and finding a job. When you need advice or want to borrow $5 or$10, whom do you turn to? The answer is undoubtedly certain members of your networks—your friends, family, and so forth.
The indirect links you have to people through your networks can help you find a job or even receive better medical care. For example, if you come down with a serious condition such as cancer, you would probably first talk with your primary care physician, who would refer you to one or more specialists whom you do not know and who have no connections to you through other people you know. That is, they are not part of your network. Because the specialists do not know you and do not know anyone else who knows you, they are likely to treat you very professionally, which means, for better or worse, impersonally.
Now suppose you have some nearby friends or relatives who are physicians. Because of their connections with other nearby physicians, they can recommend certain specialists to you and perhaps even get you an earlier appointment than your primary physician could. Because these specialists realize you know physicians they know, they may treat you more personally than otherwise. In the long run, you may well get better medical care from your network through the physicians you know. People lucky enough to have such connections may thus be better off medically than people who do not.
But let’s look at this last sentence. What kinds of people have such connections? What kinds of people have friends or relatives who are physicians? All other things being equal, if you had two people standing before you, one employed as a vice president in a large corporation and the other working part-time at a fast-food restaurant, which person do you think would be more likely to know a physician or two personally? Your answer is probably the corporate vice president. The point is that factors such as our social class and occupational status, our race and ethnicity, and our gender affect how likely we are to be involved in networks that can help us get jobs, good medical care, and other advantages. As just one example, a study of three working-class neighborhoods in New York City—one white, one African American, and one Latino—found that white youths were more involved through their parents and peers in job referral networks than youths in the other two neighborhoods and thus were better able to find jobs, even if they had been arrested for delinquency (Sullivan, 1989). Sullivan, M. (1989). Getting paid: Youth crime and work in the inner city. Ithaca, NY: Cornell University Press. This study suggests that even if we look at people of different races and ethnicities in roughly the same social class, whites have an advantage over people of color in the employment world.
A network is the totality of relationships that link us to other people and groups and through them to still other people and groups. Our involvement in certain networks can bring certain advantages, including better medical care if one’s network includes a physician or two.
Gender also matters in the employment world. In many businesses, there still exists an “old boys’ network,” in which male executives with job openings hear about male applicants from male colleagues and friends. Male employees already on the job tend to spend more social time with their male bosses than do their female counterparts. These related processes make it more difficult for females than for males to be hired and promoted (Barreto, Ryan, & Schmitt, 2009). Barreto, M., Ryan, M. K., & Schmitt, M. T. (Eds.). (2009). The glass ceiling in the 21st century: Understanding barriers to gender equality. Washington, DC: American Psychological Association. To counter these effects and to help support each other, some women form networks where they meet, talk about mutual problems, and discuss ways of dealing with these problems. An example of such a network is The Links, Inc., a community service group of 12,000 professional African American women whose name underscores the importance of networking. Its members participate in 270 chapters in 42 states, Washington, DC, and the Bahamas. Every 2 years, more than 2,000 Links members convene for a national assembly at which they network, discuss the problems they face as professional women of color, and consider fund-raising strategies for the causes they support.
Key Takeaways
• Groups are a key building block of social life but can also have negative consequences.
• Primary groups are generally small and include intimate relationships, while secondary groups are larger and more impersonal.
• Reference groups provide a standard for guiding and evaluating our attitudes and behaviors.
• Social networks are increasingly important in modern life and involvement in such networks may have favorable consequences for many aspects of one’s life.
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# Can circularly polarized light interfere with linearly polarized
1. ### Albert V
26
Can circularly polarized light interfere with linearly polarized light?
2. ### chrisbaird
614
Re: Interference
Yes. You can think of circularly polarized light as the coherent sum of two orthogonal, linearly polarized waves that are 90 degrees out of phase. So the other linearly polarized wave will interfere with one of the components.
3. ### clem
1,276
Re: Interference
The two beams would have to come from splitting single beam.
4. ### y33t
107
Re: Interference
Why ?
5. ### clem
1,276
Re: Interference
The two interfering beams must be coherent. Two independent beams would have incoherent phase relations.
6. ### y33t
107
Re: Interference
They can be coherent and belong to different identical sources ?
7. ### Claude Bile
1,479
Re: Interference
Coherence is a bit of a red herring here. Do the Jones calculus and you can see quite clearly that a CP wave and a LP wave can interfere (or indeed, must interfere if they overlap spatially).
Claude.
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# Grammar of words with exactly $k$ prefixes in another grammar
Given a context-free grammar $$G$$, how can one systematically construct a grammar $$G_k$$ such that
$$L(G_k) = \{w \in \Sigma^* : |\text{Pref}(w) \cap L(G)| = k\}$$
where $$\text{Pref}(w)$$ is the set of $$|w|+1$$ prefixes of $$w$$? Assume $$G$$ is unambiguous, if needed.
Consider the language $$\{ a^nb^n \mid n\ge 1\} \cup \{ a^kb^nc^n \mid k,n\ge 1 \}$$.
The language that has two prefixes in this language is $$\{ a^nb^nc^n \mid n\ge 1\}\cdot \{a,b,c\}^*$$, which is not context-free.
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# FFT based algorithm for special matrices
Contest problems with connections to deeper mathematics.
This question is with regard to Elkies' answer to the above post.
Vandermonde determinant can be computed using FFT techniques.
Can Moore determinant(including modulo some integer) be computed using FFT techniques?
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## 1 Answer
Not exactly my field, but I think that the same techniques used for Vandermonde's determinant work for Moore's and on finite rings (with the added complication of using FFTs over rings).
A good reference is Bini and Pan, Polynomial and matrix computations.
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# Math Help - Question about presentation of cyclic group
1. ## Question about presentation of cyclic group
I want to list the elements of $T=$
So from my understanding, $a^ia^j=a^{i+j} \ \ \ \forall 0 \leq i,j \leq 5$ with $a^{i+j}=a^{i+j-6} \ \ i+j \geq 6$
But I'm having trouble trying to make senses of the two other conditions, and how to properly translate them into the table.
Thank you!!!
2. Notice first that $b^4 = (b^2)^2 = (a^3)^2 = a^6 = 1$.
From the relation $bab^{-1} = a^{-1}$ it follows that $ba = a^{-1}b = a^5b$. Using that, in any product of a's and b's you can always push the a's to the left of the b's. For example $aba^2b = a(ba)ab = a(a^5b)ab = bab = a^5b^2$. In that way, you can express any element of the group as $a^ib^j$, with $0\leqslant i\leqslant5$ and $0\leqslant j\leqslant3$. That gives you 24 elements of the group, and you still have a fair amount of work to do if you want to write down the whole multiplication table.
3. and to complete Opalg's argument : also note that since $b^2=a^3$ and $b^3=a^3b,$ every element of $T$ can be written as $a^ib^j$ with $0 \leq i \leq 5$ and $0 \leq j \leq 1.$ it's easy to show that this
presentation is unique. so $T$ is a group of order $12$ with elements $1,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b.$ the group $T$ is called dicyclic group.
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# Depreciation
In depreciation, plant asset’s cost allocating over the period of assets. This process included the asset’s allocation, expenses against revenue over the asset’s life.
Depreciation methods
Straight line
Unit of production
Declining balance
Sum of year-digits
## Straight-line method
In the straight-line method, In each year of assets use, an equal amount of depreciation assigned. Depreciable cost is divided by useful life in years to the expense.
Straight-line depreciation per year=$\frac{Cost-Residual&space;value}{useful&space;life&space;in&space;years}$
=$\frac{41000-1000}{5}$
=8000
### A unit of production method
In the unit of production(UOP) method. In the plant assets, a fixed amount of depreciation charged. Depreciable cost is divided by useful life in units to determine this amount. This per-unit expense is multiplied by the number of units produced each period to compute for the period.n
Units-of-production per unit of output =$\frac{Cost-residual&space;value}{Usefullife&space;in&space;units}$
=$\frac{41000-1000}{400,000&space;miles}$
=\$.10
##### Double-Declining balance(DDB) method
– In Double-declining balance(DDB), a longer amount of asset’s cost of its useful life writes off than does straight-line, DDB multiplying the asset’s book value, through a constant percentage, which is two times the straight-line depreciation rate.DDB amounts computed as follows.
First, the straight-line depreciation rate per year computed.
Second, the straight-line rate multiplied by 2 to compute the DDB rate
Third, The DDB rate multiplied by the periods beginning asset book value cost less accumulated depreciation. The residual value of the asset ignored in computing depreciation by the DDB method except during the last year.
DDB rate per year=$(\frac{1}{useful&space;life&space;in&space;a&space;year}\times&space;2)=\frac{1}{5&space;years}\times&space;2$
=$(20%\times&space;2)=40%$
Fourth, the final year’s amount needed to reduce the assets’ book value.
4)
###### Sum-of-year-Digits(Syd) method
In the sum-of-years digits(Syd) method another accelerated method of depreciation figured by multiplying the depreciable cost of the asset by a fraction is the sum of years’ digit for a 5-year asset the year’s digits are 1,2,3,4 and 5 and their sum is 15(1+2+3+4+5=15)
Sum of the years’digits=N(N+1)/2
Where N is the useful life of the asset expressed in years, for example, when N equals 5, we have $5\frac{(5+1)}{2}=\frac{30}{2}=15$
The numeration of the SYD fraction for the first year of a 5 year asset is 5, The numerator is 4 for the third year,2 for the fourth year, and 1 for the fifth year.
The SYD equation for the limit.
SYD depreciation per year =$(Cost-Residual&space;value)\times&space;\frac{Years'digits,largest&space;first}{Sum&space;of&space;years'digits}$
=$(41000-1000)\times&space;\frac{^{5\bigstar&space;}}{1+2+3+4+5}$
=$40000\times&space;\frac{5}{15}=13,333$
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# Is it possible to have a sequence that is strictly increasing but whose first term is the maximum?
Edit 1: Changed from monotonically increasing sequence to a strictly increasing sequence.
Edit 2: Let $$\{a_n\}$$ denote a sequence such that for all $$n\geq 2$$, $$a_n < a_{n+1}$$ and $$a_n \leq a_1$$.
Is it possible to have a sequence that is strictly increasing but whose first term is the maximum?
I think not.
Let $$\{a_n\}$$ be a sequence such that for all $$n\geq 2$$ we have $$a_n < a_{n+1}$$ Suppose that $$a_n \leq a_1$$ for all $$n\geq 2$$. Then $$\{a_n\}$$ is an infinite increasing sequence that has a maximum, so $$\{a_n\}$$ is finite, which is a contradiction.
• so {a_n} is finite, which is a contradiction Why would that be a contradiction? – dxiv Sep 26 '18 at 0:25
• $\{0,0,\cdots \}$ is such a sequence. – Kabo Murphy Sep 26 '18 at 0:26
• Because $\{a_n\}$ is a sequence, which is a function whose domain is the set of natural numbers. Thus there are as many terms in the sequence as there are natural numbers, so $\{a_n\}$ is an infinite sequence. But I've shown that $\{a_n\}$ is finite, which is a contradiction. – Alana Sep 26 '18 at 0:26
• @Alana Take any strictly increasing convergent sequence $a_n\to A$, pick a $B \ge A$, then replace $a_1$ with $B$. – dxiv Sep 26 '18 at 0:58
• A note on why you must pick a sequence which converges $a_n\to A$ is due to the monotone convergence theorem – JMoravitz Sep 26 '18 at 1:00
As Kavi Rama Murphy pointed out in his comment, $$a_n = 0$$ satisfies the criterion you described. However,it is true that there exists no strictly increasing sequence with more than a single element for which $$a_0 = \text{max}\{a_n\}$$ is the maximum. In particular, because $$a$$ is strictly increasing, $$a_0 < a_1$$, which implies that $$\text{max}\{a_n\}_{n \leq 1} = a_1 > a_0$$. Further, since $$\{a_n\}_{n \leq 1} \subset \{a_n\}$$, we have $$\text{max}\{a_n\} \geq \text{max}\{a_n\}_{n \leq 1} = a_1 > a_0$$.
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## Deviations from the mean of a sum of independent, non-identically-distributed Bernoulli variables
### September 29, 2011
Let Bi, i = 1, …, n be independent Bernoulli variables with parameters qi, i = 1, …, n, respectively. Let S be their sum. For convenience, assume q1 ≥ q2 ≥ ··· ≥ qn. I wish to bound tightly from above the probability that S is greater or equal to some l, having the bound depend solely on ES = q1 + ··· + qn.
Clearly, if l ≤ ES, then the tightest bound is 1. This is attained by setting q1 = ··· = ql = 1.
This example shows that while the variance of S is maximized by setting qi = ES / n, i = 1, ···, n, at least for some values, l, P(S ≥ l) is maximized by having the Bi not identically distributed.
##### Proposition 1:
For every l, P(S ≥ l) is maximized when q1 = ··· = qmo = 1, qmo+1 = ··· = qn-mz = q, and qn-mz+1 = ··· = qn = 0, for some mo and mz, and for q = (ES – mo) / (n – mo – mz).
##### Proof:
Assume that 1 > qi > qj > 0. Let S’ = S – Bi – Bj. Then
P(S ≥ l) = P(S’ ≥ l) + p1 (qi + qj – qi qj) + p2 qi qj,
where p1 = P(S’ = l – 1) and p2 = P(S’ = l – 2). Thus, keeping qi + qj fixed, but varying the proportion between them, P(S ≥ l) is a linear function of qi qj. Unless p1 = p2, P(S ≥ l) will be increased by varying qi and qj – with a maximum either when they are equal or when one of them is zero or one.
Therefore, P(S ≥ l) cannot be at a maximum if there exist 1 > qi > qj > 0, unless p1 = p2. But in that case, the same probability can be achieved by setting the parameter of Bi to be q’i = 0 (if qi + qj < 1) or q’i = 1 (otherwise), and the parameter of Bj to be q’j = qi + qj – q’i. Therefore, in that case, there would exist a set of parameters, q’1, …, q’n, that would achieve that same probability but with fewer parameters that are not equal to zero or one. Thus, in the set of parameter settings maximizing P(S ≥ l), there exists a solution – namely the one which maximizes the number of parameters with extreme values (zeros and ones) – in which there is only one non-extreme value. ¤
The next step is to investigate which specific parameter setting correspond to various combinations of l and ES.
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# Find a number such that the sum of the number and its square will be as small as possible.
Find a number such that the sum of the number and its square will be as small as possible.
Please try to explain as well as you can.
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Hint: You are trying to find the minimum of $x^2+x$. Try expressing this as a completed square: $(x+1/2)^2-1/4$ and see what you can conclude. – Old John Jun 10 '12 at 21:55
Why have you tagged this trigonometry? I have removed the trigonometry tag and retagged it as algebra-precalculus. – user17762 Jun 10 '12 at 21:56
Apologies, @Marvis. I suggested in a previous post that the OP also tag what class the homework was for, to give us so.e context. – Cameron Buie Jun 10 '12 at 22:58
Let the number be $x$. Then you want to find $x$ such that $x+x^2$ is minimum.
Note that you can write $x+x^2$ as $\left( x+ \dfrac12\right)^2 - \dfrac14$.
Can you finish it from here, by noting that a square term is always non-negative?
Move your mouse over the gray area to see the complete answer.
As stated above, a square term is always non-negative. Hence, we have that $\left( x+ \dfrac12\right)^2 \geq 0$. Adding $-\dfrac14$ to both sides, we get that $x + x^2 = \left( x+ \dfrac12\right)^2 - \dfrac14 \geq -\dfrac14$. Hence, the minimum value is $-\dfrac14$ and is attained when $\left( x+ \dfrac12\right)^2 = 0$ i.e. when $\left( x+ \dfrac12\right) = 0$. Hence the minimum value is $-\dfrac14$ and is attained at $x = -\dfrac12$.
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We want to find the number $x$ that minimizes $x^2 + x$, the derivative of which is $2x + 1$. To find critical points, we solve $2x+1 = 0$ to get $x = -1/2$.
So far, we have shown only that $x = 1/2$ is either a local maximum or a local minimum of $x^2 + x$. Considering the shape of the graph, however, we know that it is a parabola opening upward, and so has no local maxima. Thus, $x = -1/2$ is our desired minimum value.
(Another way to distinguish between local maxima and minima is to consider the second derivative, which is the constant function $2$ in this case. Since this is positive for all $x$, the graph of $x^2 + x$ is concave up for all $x$. Thus, we again conclude that $x = -1/2$ must be a local minimum.)
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# Caption This: The Bell Tolls for Whitney
Enter your caption for a chance to be featured in our weekly newsletter.
Caption this photo!
What's Whitney Sudler-Smith doing here? You tell us. Write a caption for thisphoto. If yours is chosen as the best, it'll appear in Bravo's e-mail newsletter next Thursday.
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# Elemental Signals
Elemental signals are the building blocks with which we build complicated signals. By definition, elemental signals have a simple structure. Exactly what we mean by the "structure of a signal" will unfold in this section of the course. Signals are nothing more than functions defined with respect to some independent variable, which we take to be time for the most part. Very interesting signals are not functions solely of time; one great example of which is an image. For it, the independent variables are
and
(two-dimensional space). Video signals are functions of three variables: two spatial dimensions and time. Fortunately, most of the ideas underlying modern signal theory can be exemplified with one-dimensional signals.
## Sinusoids
Perhaps the most common real-valued signal is the sinusoid.
For this signal,
is its amplitude,
its frequency, and
its phase.
## Complex Exponentials
The most important signal is complex-valued, the complex exponential.
Here, i denotes
.
is known as the signal's complex amplitude. Considering the complex amplitude as a complex number in polar form, its magnitude is the amplitude
and its angle the signal phase. The complex amplitude is also known as a phasor. The complex exponential cannot be further decomposed into more elemental signals, and is the most important signal in electrical engineering! Mathematical manipulations at first appear to be more difficult because complex-valued numbers are introduced. In fact, early in the twentieth century, mathematicians thought engineers would not be sufficiently sophisticated to handle complex exponentials even though they greatly simplified solving circuit problems. Steinmetz introduced complex exponentials to electrical engineering, and demonstrated that "mere" engineers could use them to good effect and even obtain right answers! See Complex Numbers for a review of complex numbers and complex arithmetic.
The complex exponential defines the notion of frequency: it is the only signal that contains only one frequency component. The sinusoid consists of two frequency components: one at the frequency
and the other at
.
### Euler relation:
This decomposition of the sinusoid can be traced to Euler's relation.
### Decomposition:
The complex exponential signal can thus be written in terms of its real and imaginary parts using Euler's relation. Thus, sinusoidal signals can be expressed as either the real or the imaginary part of a complex exponential signal, the choice depending on whether cosine or sine phase is needed, or as the sum of two complex exponentials. These two decompositions are mathematically equivalent to each other.
Using the complex plane, we can envision the complex exponential's temporal variations as seen in the above figure (Figure). The magnitude of the complex exponential is
, and the initial value of the complex exponential at
has an angle of
. As time increases, the locus of points traced by the complex exponential is a circle (it has constant magnitude of
). The number of times per second we go around the circle equals the frequency
. The time taken for the complex exponential to go around the circle once is known as its period
, and equals
. The projections onto the real and imaginary axes of the rotating vector representing the complex exponential signal are the cosine and sine signal of Euler's relation (Equation).
## Real Exponentials
As opposed to complex exponentials which oscillate, real exponentials decay.
The quantity
is known as the exponential's time constant, and corresponds to the time required for the exponential to decrease by a factor of
, which approximately equals 0.3680.368. A decaying complex exponential is the product of a real and a complex exponential.
In the complex plane, this signal corresponds to an exponential spiral. For such signals, we can define complex frequency as the quantity multiplying
.
## Unit Step
The unit step function is denoted by
, and is defined to be
### Origin warning:
This signal is discontinuous at the origin. Its value at the origin need not be defined, and doesn't matter in signal theory.
This kind of signal is used to describe signals that "turn on" suddenly. For example, to mathematically represent turning on an oscillator, we can write it as the product of a sinusoid and a step:
.
## Pulse
The unit pulse describes turning a unit-amplitude signal on for a duration of
seconds, then turning it off.
We will find that this is the second most important signal in communications.
## Square Wave
The square wave
is a periodic signal like the sinusoid. It too has an amplitude and a period, which must be specified to characterize the signal. We find subsequently that the sine wave is a simpler signal than the square wave.
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# How does the viscosity (besides density) of a fluid affects sinking
If water, air of whatever fluid had a different viscosity, but the same density, would things fall/sink differently?
• [Related Quora Would-a-hydrophobic-coating-make-a-heavy-object-sink-through-water-faster] (quora.com/Science/…) – user28737 Jun 4 '14 at 6:07
$$\frac{2(\rho_s -\rho_f)}{9\mu}gR^2$$
where $\rho_s$ and $\rho_f$ are the densities of the sphere, $\mu$ is the dynamic viscosity, $g$ is the acceleration due to gravity, and $R$ is the radius of the sphere.
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Create new environment with options
I'm new to creating new environments and even though I searched at the xparse*documentation I couldn't figured out how to solve my specific problem. I would like to create a new environment, say concept, and I would like it to take an optional argument (I don't know if it has to be a boolean or what). This optional argument will make the text to be in italics. If the argument is not declared, it will simply print normal font. It's as simple as that. So
\begin{concept}[i] % I just invented a option named "i"
This text will be in italics
\end{concept}
or without an option
\begin{concept}
This text will be in normal font.
\end{concept}
I thought that may be using the xparse package would make the task easier. Sorry to ask such a silly question but I couldn't come to a solution.
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For simple usages see Peter's answer. For advanced stuff, try some of the keyval packages like kvoptions or pgfkeys, xkeyval, l3keys etc. – tohecz Feb 20 at 13:31
Welcome to TeX.sx! Your post was migrated here from Stack Overflow site. Please register on this site, too, and make sure that both accounts are associated with each other (by using the same OpenID), otherwise you won't be able to comment on or accept answers or edit your question. – Werner Feb 20 at 15:21
migrated from stackoverflow.comFeb 20 at 13:17
This question came from our site for professional and enthusiast programmers.
Normal LaTeX environments can do this without the need for the xparse package:
References:
Relevant for using \itshape:
Code:
\documentclass{article}
\usepackage{xstring}
\newenvironment{concept}[1][]{%
\IfStrEq{#1}{i}{\itshape}{}%
}{%
}%
\begin{document}
\begin{concept}[i]% I just invented a option named "i"
This text will be in italics
\end{concept}
\begin{concept}
This text will be in normal font.
\end{concept}
\end{document}
-
Thanks @peter ! – ezitoc Feb 19 at 19:11
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# First-order properties of Euclidean fields (instead of real closed fields)
Let $$\mathbb{F}$$ be an ordered field.
• $$\mathbb{F}$$ is called Euclidean if $$\forall x>0 \in \mathbb{F}\ \exists y \in \mathbb{F} : y^2=x$$, i.e. a square root exists.
• $$\mathbb{F}$$ is called real closed if it is Euclidean and every polynomial of odd degree has at least one zero in $$\mathbb{F}$$.
Real closed fields have the same first-order properties as the field of real numbers, i.e. statements that involve symbols like $$+, \cdot, =,\leq,\dots$$ are true for a real closed field $$\mathbb{F}$$ if and only if they are true over the real numbers $$\mathbb{R}$$.
If real closed fields are generalisations of the real numbers, then the Euclidean fields are generalisations of the field of constructible numbers. It is wrong to say that Euclidean fields have the same first-order properties as the constructible numbers, since all real closed fields are Euclidean, so the property of having a $$n$$th root may be different. But is the following true?
A first-order-logic statement holds over the constructible numbers, but not over the real numbers, if and only if it holds for any Euclidean field $$\mathbb{F}$$, but not its real closure $$\overline{\mathbb{F}}$$.
Is there a counter example? Does one direction hold? Are there references to this? More generally, is there any way in which real closed fields are better behaved under "logical generalisations" than the Euclidean fields?
I am interested in this question because in my research the Euclidean property seems to suffice most of the time, but people just use real closed fields since they have a well established theory.
• I think your forward direction is trivially false. Constructible number don't have cube root of 2, but Euclidean field can have cube root of 2, simply by constructing an Euclidean field over $Q(2^{1/3})$. As for the backward direction, what does it mean to hold over all Euclidean field but not real closure? Real closed field are Euclidean. – needanstyvm Oct 28 '19 at 13:50
• Thanks, you are right. Having a cube root, having a fifth root, having a severth root etc. are all statements that are not true for the constructible numbers, but they are true for some Euclidean non real closed fields. I wonder if these examples are "the only" ones. Backward direction: There might be a property in a Euclidean $\mathbb{F}$, witch does not hold in it's real closure $\overline{\mathbb{F}}$. – Strichcoder Oct 28 '19 at 14:16
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## Monday, March 27, 2017
### Week of March 27,2017
Algebra 1
A Quadratic Equation in Standard Form
(ab, and c can have any value, except that a can't be 0.)
To "Factor" (or "Factorise" in the UK) a Quadratic is to:
find what to multiply to get the Quadratic
It is called "Factoring" because we find the factors (a factor is something we multiply by)
## Example
The factors of x2 + 3x − 4 are:
(x+4) and (x−1)
Why? Well, let us multiply them to see:
(x+4)(x−1) = x(x−1) + 4(x−1) = x2 − x + 4x − 4 = x2 + 3x − 4
Multiplying (x+4)(x−1) together is called Expanding.
In fact, Expanding and Factoring are opposites:
Expanding is easy, but Factoring can often be tricky
It is like trying to find out what ingredientswent into a cake to make it so delicious. It can be hard to figure out!
So let us try an example where we don't know the factors yet:
## Common Factor
First check if there any common factors.
### Example: what are the factors of 6x2 − 2x = 0 ?
6 and 2 have a common factor of 2:
2(3x2 − x) = 0
And x2 and x have a common factor of x:
2x(3x − 1) = 0
And we have done it! The factors are 2x and 3x − 1,
We can now also find the roots (where it equals zero):
• 2x is 0 when x = 0
• 3x − 1 is zero when x = 1/3
And this is the graph (see how it is zero at x=0 and x=1/3):
But it is not always that easy ...
## Guess and Check
Maybe we can guess an answer?
### Example: what are the factors of 2x2 + 7x + 3 ?
No common factors.
Let us try to guess an answer, and then check if we are right ... we might get lucky!
We could guess (2x+3)(x+1):
(2x+3)(x+1) = 2x2 + 2x + 3x + 3
= 2x2 + 5x + 3
(WRONG)
(2x+7)(x−1) = 2x2 − 2x + 7x − 7
= 2x2 + 5x − 7
(WRONG AGAIN)
OK, how about (2x+9)(x−1):
(2x+9)(x−1) = 2x2 − 2x + 9x − 9
= 2x2 + 7x − 9
(WRONG AGAIN)
Oh No! We could be guessing for a long time before we get lucky.
That is not a very good method. So let us try something else.
## A Method For Simple Cases
Luckily there is a method that works in simple cases.
With the quadratic equation in this form:
Step 1: Find two numbers that multiply to give ac (in other words a times c), and add to give b.
Example: 2x2 + 7x + 3
ac is 2×3 = 6 and b is 7
So we want two numbers that multiply together to make 6, and add up to 7
In fact 6 and 1 do that (6×1=6, and 6+1=7)
How do we find 6 and 1?
It helps to list the factors of ac=6, and then try adding some to get b=7.
Factors of 6 include 1, 2, 3 and 6.
Aha! 1 and 6 add to 7, and 6×1=6.
Step 2: Rewrite the middle with those numbers:
Rewrite 7x with 6x and 1x:
2x2 + 6x + x + 3
Step 3: Factor the first two and last two terms separately:
The first two terms 2x2 + 6x factor into 2x(x+3)
The last two terms x+3 don't actually change in this case
So we get:
2x(x+3) + (x+3)
Step 4: If we've done this correctly, our two new terms should have a clearly visible common factor.
In this case we can see that (x+3) is common to both terms, so we can go:
Start with 2x(x+3) + (x+3) Which is: 2x(x+3) + 1(x+3) And so: (2x+1)(x+3)
Check: (2x+1)(x+3) = 2x2 + 6x + x + 3 = 2x2 + 7x + 3 (Yes)
Algebra 2
Assessments: Midterm on Wednesday
Monday
Today, we look at multiply and divide rational expressions
Help:http://www.purplemath.com/modules/rtnlmult.htm
HW; None
Tuesday
Combo day, ( simplify, multiply and divide) and rational equations
HW: review for midterm
Wednesday
Midterm day and classwork time on rational equations
Help:http://www.purplemath.com/modules/solvrtnl.htm
HW: None
Thursday
A quick quiz today on the rational expressions and notes on word problems for rational equations
HW: None
Friday
Make up for the 2nd 6 weeks, find what you need to make up and do it. If all work is done, challenge puzzles will be provided.
HW; none ( enjoy the break!)
Enjoy Spring Break! 6.5 weeks to go!
## Monday, March 20, 2017
### Week of March 20,2017
ALGEBRA 1
Choosing a Factoring Method
Step 1 Check for a greatest common factor.
Step 2 Check for a pattern that fits the difference of two squares or a perfect-square trinomial.
Step 3 To factor x 2 + bx + c, look for two numbers whose sum is b and whose product is c. To factor ax 2 + bx + c, check factors of a and factors of c in the binomial factors. The sum of the products of the outer and inner terms should be b.
Step 4 Check for common factors.
Use the following table to help you choose a factoring method.
First factor out a GCF if possible.
Then, Explain how to choose a factoring method for x2 − x − 30. Then state the method. • There is no GCF. • x2 − x − 30 is a trinomial.
• The terms a and b are not perfect squares, therefore this is not a perfect square trinomial. • a = 1 Method: Factor by checking factors of c that sum to b. Explain how to choose a factoring method for 2x2 − 50. Then state the method.
• Factor out the GCF: 2(x2 − 25) • x2 − 25 is a binomial. • a and b are perfect squares. This is a difference of squares.
If binomial, check for difference of squares. yes no Use (a + b)(a − b). If no GCF, it cannot be factored. If trinomial, check for perfect square trinomial. yes no Factor using (a + b) 2 or (a − b) 2 . If a = 1, check factors of c that sum to b. If a ≠ 1, check inner plus outer factors of a and c that sum to b. If 4 or more terms, Try to factor by grouping.
ALGEBRA 2
Assessments: Unit project on Thursday
Monday
Today, we look at characteristics of polynomials ( intercepts, roots, extrema)
Help:http://www.coolmath.com/precalculus-review-calculus-intro/precalculus-algebra/12-relative-extrema-minimums-maximums-01
HW; None
Tuesday
It is a day of review for the characteristics of Polynomials and finding roots of polynomials, a full day all about polynomials.
Help:https://www.sophia.org/tutorials/characteristics-of-polynomials
HW; None
Wednesday
It is a day of review of all things polynomials and a look at end behavior, with discovery mixed in.
HW; Prepare for a unit test if you chose the unit test for the assessment
Thursday
Your choice, you can work on a project for the unit or take a unit test on the unit. Both items can count as a test grade.
Help:http://www.sosmath.com/algebra/factor/fac02/fac02.html
HW; None
Friday
Yes, it is Friday and we look at rational expressions, with a focus on factoring polynomials.
HW; None
Have a great weekend!
One week to Spring Break!
## Monday, March 13, 2017
Algebra 1
### Formula For Factoring Trinomials(when a =1)
It's always easier to understand a new concept by looking at a specific example so you might want to do that first. This formula works when 'a' is 1. In other words, we will use this approach whenever the coefficient in from of x2 is 1. (If you need help factoring trinomials when $a\ne 1$, then go here.
1. identify a,b, and c in the trinomial ax2 + bx+c
2. write down all factor pairs of c
3. identify which factor pair from the previous step sums up to b
4. Substitute factor pairs into two binomials
### Example of Factoring a Trinomial
##### Example 1
Factor x2 + 5x + 4
Step 1
Identify a,b, and c in the trinomial
ax2 + bx+c
a= 1
b= 5
c= 4
Step 2
Write down all factor pairs of 4
(Note: since 5 is positive we only need to think about pairs that are either both positive or both negative. Remember a negative times a negative is a positive. As the chart on the right shows you -2*-2 is positive 4...so we do have to consider these two negative factors. This is probably easier to understand if you watch our video lesson factoring trinomials)
Step 3
identify which factor pair from the previous step sums up to c
Step 4
Substitute that factor pair into two binomials
(x +4)(x+1)
Step 5
If you'd like, you can check your work by multiplying the two binomials and verify that you get the original trinomial
Algebra 2
Assessments: Unit Project given later in the week
Monday
It is the day to write equation of the polynomial from the roots and to solve polynomials with imaginary roots and real roots combined.
Help:https://www.varsitytutors.com/hotmath/hotmath_help/topics/polynomials-with-complex-roots
HW: none
Tuesday
Happy Pi Day! We will solve circle problems all day today! The work will be polynomial roots and writing equations using the roots.
Help: http://www.piday.org/
HW; Prepare for a mini quiz, a circle problem!
Wednesday
We confirm circle problems are good and there will be a mini quiz today. You will do a great job!
Help:http://www.purplemath.com/modules/fromzero2.htm
HW; None
Thursday
It is the day to look at characteristics of polynomials
HW: None
Friday
Happy St. Patrick's Day! We continue with characteristics today
Help:http://www.sosmath.com/algebra/factor/fac03/fac03.html
HW: None
Have a great weekend! Two weeks to Spring Break!
# Factoring A GCF From an Expression Lessons
To best understand this lesson, you should make sure you know how to find the GCF of two or more terms. To learn how, see the lesson called Finding a GCF.
3x3 + 27x2 + 9x
To factor out the GCF in an expression like the one above, first find the GCF of all of the expression's terms.
3 (1, 3) 27 (1, 3, 9, 27) 9 (1, 3, 9)
GCF = 3x
Next, write the GCF on the left of a set of parentheses:
3x( )
Next, divide each term from the original expression (3x3+27x2+9x ) by the GCF (3x), then write it in the parenthesis.
3x3 / 3x = x2
27x2 / 3x = 9x
9x / 3x = 3
3x(x2 + 9x + 3)
The next expression we will be factoring is shown below.
36x2 - 64y4
To begin factoring the GCF out of the expression, find the GCF of the two terms.
36 (1, 2, 3, 4, 6, 9, 12, 18, 36) 64 (1, 2, 4, 8, 16, 32, 64)
GCF = 4
As you can see, the two terms to do not have any variables in common, therefore the GCF is simply 4.
Now write 4, the GCF, on the left of a set of parentheses.
4( )
Now divide each term 4, the GCF, and place the result inside the parentheses.
36x2 / 4 = 9x2
-64y4 / 4 = -16y4
4(9x2 - 16y4)
ALGEBRA 2
Assessments: Quiz on Wednesday and Summative assessment next week
Monday
Hope you had a great weekend! Today, we are working on some polynomial cards outside the classroom with varying questions on roots and factors. Yes, you will think on this activity.
I have two colors posted, so ask me which color is the best fit for you today. Also, you find roots of 4th degree polynomials and missing terms polynomials.
Help:http://math.stackexchange.com/questions/12787/finding-roots-of-the-fourth-degree-polynomial-2x4-3x3-11x2-9x-15
HW; None
Tuesday
Today, we look at the Rational Root Theorem and find all roots of a given polynomial. I will have practice time for you today to master this concept and you will have choices on the problems.
HW; Study for quiz
Wednesday
Moving along the week, you shall shine on the formative assessment (a quiz) in class today. I will have a quiz preview first to ensure you are ready this quiz. Also, we analyze the long division process to see a connection with the polynomial, the radical and the factors.
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Using the result of Exercise 13, it is possible to avoid the discontinuity at d = 0 in the direction
Using the result of Exercise 13, it is possible to avoid the discontinuity at d = 0 in the direction finding mapping of the simple gradient projection method. At a given point let γ = − min{0, λi}, with the minimum taken with respect to the indices i corresponding the active inequalities. The direction to be taken at this point is d = −Pg if |Pg| γ , or d, defined by dropping the inequality i for which λi = −γ , if |Pg| γ . (In case of equality either direction is selected.) Show that this direction finding map is closed over a region where the set of active inequalities does not change.
Plagiarism Checker
Submit your documents and get free Plagiarism report
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# Constructing scale drawings
An urban planner needs your help in creating a scale drawing. Let's use our knowledge about scale factor, length, and area to assist.
### Problem
Rishi is making a map of a remote village. In this village, each hut has a rectangular base measuring 4, space, m by 6, space, m.
Move the points on the grid below to draw the base of one of the huts described above.
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# sum_(n=0)^oo 5^n/(3^n +4^n). Does the series converge or diverge?
## I used Divergence Test and I'm stuck on this part ${\lim}_{n \to \infty} {\left(\frac{5}{4}\right)}^{n} / \left[{\left(\frac{3}{4}\right)}^{n} + 1\right]$ Im having trouble finding the limit
Apr 9, 2017
${\sum}_{n = 0}^{\infty} {5}^{n} / \left({3}^{n} + {4}^{n}\right) = + \infty$
#### Explanation:
You are on the good path:
${5}^{n} / \left({3}^{n} + {4}^{n}\right) = {\left(\frac{5}{4}\right)}^{n} / \left(1 + {\left(\frac{3}{4}\right)}^{n}\right)$
Now as:
${\lim}_{n \to \infty} \frac{1}{1 + {\left(\frac{3}{4}\right)}^{n}} = 1$
we have:
${\lim}_{n \to \infty} {\left(\frac{5}{4}\right)}^{n} / \left(1 + {\left(\frac{3}{4}\right)}^{n}\right) = {\lim}_{n \to \infty} {\left(\frac{5}{4}\right)}^{n} \cdot {\lim}_{n \to \infty} \frac{1}{1 + {\left(\frac{3}{4}\right)}^{n}} = + \infty \cdot 1 = + \infty$
and as the sequence of the terms is not infinitesimal, the series cannot converge. As the terms are all positive the series is then divergent.
Apr 13, 2017
(I know this isn't the method you requested, but this is how I first approached the problem. With series problems, there are frequently many valid solutions.)
#### Explanation:
Note that ${3}^{n} + {4}^{n} < {4}^{n} + {4}^{n} = 2 \left({4}^{n}\right)$.
Then, ${5}^{n} / \left({3}^{n} + {4}^{n}\right) > {5}^{n} / \left(2 \left({4}^{n}\right)\right)$ since the first one has a lesser denominator.
We should see that ${\sum}_{n = 0}^{\infty} {5}^{n} / \left(2 \left({4}^{n}\right)\right) = \frac{1}{2} {\sum}_{n = 0}^{\infty} {\left(\frac{5}{4}\right)}^{n}$, which is divergent by the Geometric Series test since divergent since $\left\mid \frac{5}{4} \right\mid > 1$.
Then, by the direct comparison test, ${\sum}_{n = 0}^{\infty} {5}^{n} / \left({3}^{n} + {4}^{n}\right)$ is divergent as well since it is larger than another divergent series.
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# Sentence Examples with the word Hesperornis
The swimming Hesperornis (see Odontornithes) was also devoid of such a structure.
In Colymbus the patella is reduced to a small ossicle, its function being taken by the greatly developed pyramidal processus tibialis anterior; in Podiceps and Hesperornis the patella itself is large and pyramidal.
In length, as afterwards appeared) having some affinity, it was thought, to the Colymbidae were described under the name of Hesperornis regalis, and a few months later (iv.
View more
Two years more and the originally found Hesperornis was discovered also to have teeth, but these were inserted in a groove.
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Drawing on Canvas
Generic drawing is done on a Canvas. You control what appears on this canvas by defining a draw function:
using Gtk, Graphics
c = @GtkCanvas()
win = GtkWindow(c, "Canvas")
@guarded draw(c) do widget
ctx = getgc(c)
h = height(c)
w = width(c)
# Paint red rectangle
rectangle(ctx, 0, 0, w, h/2)
set_source_rgb(ctx, 1, 0, 0)
fill(ctx)
# Paint blue rectangle
rectangle(ctx, 0, 3h/4, w, h/4)
set_source_rgb(ctx, 0, 0, 1)
fill(ctx)
end
show(c)
This draw function will get called each time the window gets resized or otherwise needs to refresh its display.
Errors in the draw function can corrupt Gtk's internal state; if this happens, you have to quit julia and start a fresh session. To avoid this problem, the @guarded macro wraps your code in a try/catch block and prevents the corruption. It is especially useful when initially writing and debugging code. See further discussion about when @guarded is relevant.
Finally, Canvases have a field called mouse that allows you to easily write callbacks for mouse events:
c.mouse.button1press = @guarded (widget, event) -> begin
ctx = getgc(widget)
set_source_rgb(ctx, 0, 1, 0)
arc(ctx, event.x, event.y, 5, 0, 2pi)
stroke(ctx)
reveal(widget)
end
This will draw a green circle on the canvas at every mouse click. Resizing the window will make them go away; they were drawn on the canvas, but they weren't added to the draw function.
Note the use of the @guarded macro here, too.
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Free Version
Easy
# Solution to a Linear System
LINALG-G2LCXF
Consider the following system of linear equations:
$2x + 3y = 1$
$x + y = 0$
Which of the following is a solution to the system?
A
$(2,-1)$
B
$(1,-1)$
C
$(-1,1)$
D
$(0,0)$
E
$(2,3)$
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# PrusaSlicer
PrusaSlicer is Prusa Research’s slicer software, which is Open Source and a direct descendant of Slic3r (like ).
It is annoyingly ‘cyclic Open Source’, in the sense that it is developed mostly internally and the public source tree updated after each release.
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# Decidability of CFG ambiguity
I have been trying to show the following language is undecidable.
$L = \{ (\langle G \rangle , n ): G$ is a context-free grammar with an ambiguous string of length $\le n \}$.
I think it is undecidable because if $G$ doesn't have an ambiguous string, it seems like one would have to check parse trees of arbitrary heights to verify they don't yield $s$.
Or, is there some lower bound on the height of a parse tree for a given string $s$, that is any derivation with more than some number of steps cannot yield $s$? It seems like the presence of rules $S \to \varepsilon$, where $\varepsilon$ is the empty string, would ensure that there would exist parse trees of arbitrary height, in general.
• Huh? Doesn't the CYK algorithm tell you in polynomial time whether any particular string is ambiguous for a given grammar? So just try it on all short strings. Feb 28 '15 at 5:18
• CYK Assumes that the grammar is in Chomsky normal form, won't the conversion change the ambiguity? Feb 28 '15 at 7:50
• It does not matter whether one uses CYK. Any derivation of a string of length n only involves strings of length at most n and can therefore be only of length (#variables+#terminals+1)^n. The algorithm therefore only needs to cycle through derivations of exponential length. Good enough for decidability. Mar 1 '15 at 0:15
• @Shaull As I remember CNF does not change ambiguity. Anyway, you do not need CNF to apply CYK, you only need to put the grammar in binary form, wich is a trivial transformation that barely changes the parse trees. It replaces n-ary nodes by an equivalent succession of n-1 binary ones. Mar 3 '15 at 0:26
• @ThomasS You still may have to detect loops, which CYK does for you. To answer the OP, there is no upperbound on the height of a parse tree in general. Though people tend to avoid grammars that allow that. That is due to non-terminals that may derive into themselves. The empty string may help hide the phenomenon. Mar 3 '15 at 0:29
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## My Solution of The Purdue University Problem Of The Week No. 12
My solution of POW12 was accepted. [Remark of December 19, 2009: it is only stated that it was “completely or partially proved”]
Problem. Find, with proof, the maximum value of $\displaystyle\prod\limits_{j=1}^{k}x_{j}$ where $x_{j}\geq 0,\displaystyle\sum\limits_{j=1}^{k}x_{j}=100,$ and $k$ is variable. In particular, your answer must be greater than or equal to all values obtained from other choices of $k.$
Here is the solution I submited.
Solution.
Let $\left( x_{1},x_{2},\ldots ,x_{k}\right) \in\mathbb{R}^{k}$, $f\left( x_{1},x_{2},\ldots ,x_{k}\right) =\displaystyle\prod\limits_{j=1}^{k}x_{j}\in\mathbb{R}$, $c\left( x_{1},x_{2},\ldots ,x_{k}\right) =\displaystyle\sum_{j=1}^{k}x_{j}-100$ $\in\mathbb{R}$ and $\lambda\in\mathbb{R}$. For a given $k\in\mathbb{Z}$, with $k\geq 1$, we know by the Lagrange multipliers method that $f\left( x_{1}^{\ast },x_{2}^{\ast },\ldots ,x_{k}^{\ast }\right)$ is a local extremum if for $x^{\ast }=\left( x_{1}^{\ast },x_{2}^{\ast },\ldots,x_{k}^{\ast }\right)$
$\nabla f\left( x^{\ast }\right) -\nabla c\left( x^{\ast }\right) \lambda^{\ast }=0\quad \quad k\text{ equations}$
$c\left( x^{\ast }\right) =0\quad \quad 1\text{ equation}$
where $\lambda ^{\ast }$ is the value of the multiplier $\lambda$ that is a solution of these $k+1$ equations. Hence we have respectively
$\displaystyle\prod\limits_{j\neq i}^{k}x_{j}^{\ast }-\lambda ^{\ast }=0\quad\quad 1\leq i\leq k$
and
$\displaystyle\sum\limits_{j=1}^{k}x_{j}^{\ast}-100=0\text{. }$
Solving this system of equations we find
$x_{1}^{\ast }=x_{2}^{\ast }=\cdots =x_{i}^{\ast }\cdots =x_{k}^{\ast}=\lambda ^{\ast }=\dfrac{100}{k}$
and
$\displaystyle\prod\limits_{j=1}^{k}x_{j}^{\ast}=\left( \dfrac{100}{k}\right) ^{k}$,
the latter being a local extremum of $f\left( x_{1},x_{2},\ldots ,x_{k}\right)$, for a fixed $k$. We transformed the initial maximizing problem in $k$ continuous variables and the discrete variable $k$ into the maximizing of $\left( 100/k\right) ^{k}$. Now we introduce the following function
$u\left( t\right) =\left( \dfrac{100}{t}\right) ^{t}\quad\text{with }t\in\mathbb{R}\text{.}$
The function $u\left( t\right)$ has a maximum at the same point $t$ than the function
$v\left( t\right) =\ln u\left( t\right) =t\ln 100-t\ln t\text{.}$
On the other hand $v^{\prime }\left( t\right) =0$ for $t^{\ast }=e^{\ln100-1}\simeq 36.788,v^{\prime }\left( t\right) >0$ for $t and $v^{\prime }\left( t\right) <0$ for $t>t^{\ast }$. Therefore
$u\left( t^{\ast }\right) =\left( \dfrac{100}{t^{\ast }}\right) ^{t^{\ast }}$
is a maximum of $u\left( t\right)$. Since $u\left( 37\right) >u\left( 36\right)$, the maximum occurs at $k=37$. Thus, for $\sum_{j=1}^{k}x_{j}=\sum_{j=1}^{37}x_{j}=100$ we have
$\max\displaystyle\prod\limits_{j=1}^{k}x_{j}=\displaystyle\prod\limits_{j=1}^{37}x_{j}=\left( \dfrac{100}{37}\right) ^{37}\text{.}$
Update (Dec., 19,2009): some errors (identified by Rod Carvalho here) corrected.
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1. ## Rationalize the denominator
[\MATH]\frac{1}{((x)^(1/3)-(y)^(1/3))}[/MATH]
2. ## Re: Rationalize the denominator
\frac{1}{((x)^(1/3)-(y)^(1/3))}
3. ## Re: Rationalize the denominator
$\frac1{x^{1/3}-y^{1/3}}$
$=\frac1{x^{1/3}-y^{1/3}}\cdot\frac{x^{2/3}+x^{1/3}y^{1/3}+y^{2/3}}{x^{2/3}+x^{1/3}y^{1/3}+y^{2/3}}$
$=\frac{x^{2/3}+x^{1/3}y^{1/3}+y^{2/3}}{x-y}$
Note that the expression which I multiplied the numerator and denominator by is just the second half of the difference of cubes formula.
4. ## Re: Rationalize the denominator
I've noticed... but I don't understand why you have to do that but rules are rules I guess... Thank you so much for your response!!! But how is that x-y in the denominator?
5. ## Re: Rationalize the denominator
Originally Posted by purplec16
I've noticed... but I don't understand why you have to do that but rules are rules I guess... Thank you so much for your response!!! But how is that x-y in the denominator?
$\left(x^{1/3}-y^{1/3}\right)\left(x^{2/3}+x^{1/3}y^{1/3}+y^{2/3}\right)$
Distributing, we have
$=x+x^{2/3}y^{1/3}+x^{1/3}y^{2/3}-x^{2/3}y^{1/3}-x^{1/3}y^{2/3}-y$
Now, combining like terms leaves
$=x-y$.
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# How does Hanson's Market Maker (LMSR) work?
If the market maker wants to quote a "current price", he can. The current price for outcome 1 is:
$$\mbox{price1} = \frac{e^{\frac{q1}{b}}}{e^{\frac{q1}{b}} + e^{\frac{q2}{b}}}$$
Why this is the case? Is it just some simple "see-saw" algorithm? Exert pressure on one side and it will simply radiate across into the corresponding +/- price change?
I am asking in the context of writing a simple market-making algorithm to offer bids and quotes (on a virtual market), but this seems too simple.
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Is this related to your university project? – chrisaycock Apr 24 at 19:55
Its related to the algorithm I linked to in the question. – user997112 Apr 24 at 20:33
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This is convenient if your keyboard lacks some desired accents and other diacritics. The software is fast, accurate, and highly configurable. ), and shapes (these days, generally PostScript Type 1 or TrueType or OpenType). org) is a developing collection of resources for engaging students in writing and speaking about mathematics. University course notes, academic papers, theses: Computer Modern is there! We get back to the agreeableness of Baskerville above: perhaps people trust this font because trusted people use this font. It is an appropriate tool for:. Thanks for contributing an answer to Mathematics Stack Exchange! Please be sure to answer the question. Spacing in math mode; Integrals, sums and limits; Display style in math mode; List of Greek letters and math symbols; Mathematical fonts Figures and tables. Basic LaTeX commands. Find the best 2 free fonts in the Symbols, Math style. These ttf files use a non-standard encoding specifically for jsMath. GNU TeXmacs is a free scientific editing platform designed to create beautiful technical documents. Font sizes are identified by special names, the actual size is not absolute but relative to the font size declared in the \documentclass statement (see Creating a document in LaTeX). I know the default setting (Computer Modern's family) is a very good choice. This gist lets you define which symbols insert Unicode symbols, so e. How to use Tabs in LaTeX. LaTeX math. AMS-LaTeX If you're typing math, you should check out (and use) AMS-LaTeX. fonts latex mathematica-online. The \boldmath declaration switches to a bold math italic font; this causes letters, numbers, and most symbols used in math mode to be set in a bold type. The design isn't just intended for business documents: The regular weight has been extended with a large set of math and science symbols. This article shows several fonts for use in math mode. The Greek and Cyrillic has been designed under close supervision of an international team of experts, who aimed to set a historical new standard in multi-script type design. For this example, plot y = x 2 sin (x) and draw a vertical line at x = 2. In particular, this means most mathematical characters. Mathematical Bold Fraktur. Latex is so powerful, but if it makes no attempt to become more user-friendly, a more flexible and powerful version of something like Markdown will come along, and Latex will become irrelevant. In LaTeX, of course, all this stuff is automated: there is a scheme that, for each (text) font size, determines what maths font sizes are to be used. The jsMath package is designed to work best if you have installed the TeX font set. List of LaTeX mathematical symbols. , in order to get a particular bug fix. Patrick Just Math Tutorials patrickjmt. AMS to create the math font Euler for use in Concrete Mathematics. Getting started with XeLaTeX. (Doesn't work with numbers and nonalphabetical symbols. Using "properties" in Adobe reader the only fonts in the PDF are Times Roman, Times Roman Bold (probably used for the headings) and a lot of the Computer Modern fonts for the math. Latex, or , (pronounced lah-TEK or lay-TEK) is a typesetting markup language that is useful to produce properly formatted mathematical and scientific expressions. For several years now, I've been trying to get my hands on the BA Math Font from micropress. When using the \scalebox command from the graphicx package one can specify the width (or height) and the other dimension will be scaled proportionally. Text can be added to Jupyter Notebooks using Markdown cells. Department of Mathematics University of Manitoba Winnipeg, Manitoba Canada R3T 2N2 Library of Congress Cataloging-in-Publication Data Gr¨atzer, George A. To use tabs, you want to use the "tabbing" environment. It is an OpenType version of Latin Modern, a clone of DEK's Computer Modern, based on Monotype Modern, and of the AMS symbol fonts. Word-to-LaTeX is a program for converting Microsoft Word documents into LaTeX and XML formats (additional formats can be easily added through the configuration). [Serif Fonts] [Serif Fonts, Sub-Categorised] [Sans Serif Fonts] [Typewriter Fonts] [Calligraphical and Handwritten Fonts] [Uncial Fonts] [Blackletter Fonts] [Other Fonts] [Fonts with Math Support] [Fonts with OpenType or TrueType Support] [All Fonts, by category] [All Fonts, alphabetically] [About The L a T e X Font Catalogue] [Packages that. Call it $\boldsymbol\beta$. Add text to the graph that contains an integral expression using LaTeX markup. For some reason, latex doesn't really like making greek letters bold. and math fonts called hfbright [19] using mftrace. 47 KB This is XeTeXk, Version 3. LaTeX forum ⇒ Text Formatting ⇒ How to have Italic AND bold, in math mode ? Information and discussion about LaTeX's general text formatting features (e. AMS to create the math font Euler for use in Concrete Mathematics. LaTEX's encoding models. FAQ Contact 100% Free For Commercial Use. More details. LaTeX is a system for preparing documents to be printed or displayed. 5 centimeter leading. font to be replaced and the font spec (with encoding) of the font to be used as companion. LaTeX uses Computer Modern by default. Notons que LaTeX 2ε utilise un système de gestion des polices appelé NFSS (new font selection scheme), qui permet de les manipuler simplement (par exemple d'avoir du gras et italique en même temps). Enter text must be in text mode and writing mathematical in math mode. The LaTEX world of symbols. Articles must be of interest for a large audience in operations research, written in clear and correct English, and typeset in LaTeX. Beautiful Math with LaTeX LaTeX is a powerful markup language for writing complex mathematical equations, formulas, and more. Abstract: Categorification is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called coherence laws'. The Latin Modern Math (LM Math) font completes the modernization of the Computer Modern family of typefaces designed and programmed by Donald E. Now if you need to add normal text into a formula or even write a formula using words, you can do this with the text-command inside the math-environment:. Renders in the predefined maths bold font. To use LaTeX markup, set the Interpreter property for the Text object to 'latex'. Give your brain a workout!. But it isn't. Re: [solved] Type1 Latex Math Fonts It's unclear to me what font you are using. This is great for teachers or educators who use LaTex in the University including Algebra professors, Math professors but also Statistics courses, economy or financial subjects. Once you install the free addin provided by IguanaTex, you can add new equations to your PowerPoint slides just like in the example above. Email this graph HTML Text To: You will be emailed a link to your saved graph project where you can make changes and print. I found a solution for MS Word (Select a different math font in Microsoft Word). First, you need to have a font that has the math characters you need. Students, teachers, parents, and everyone can find solutions to their math problems instantly. sscript-font Fonttousefornestedsub-andsuper-scripts section§4. Full Text Search. Having done so, she is pleased to notice that Hello World is printed in the same font as the rest of her document (Computer Modern). It is here where GUST's e-foundry guys, Bogusław “Jacko” Jackowski, Janusz M. Here's how to use them. Page breaks in math environments []. 020 onward (October 2011) all fonts contain a very complete glyph set for mathematical typesetting. Just add a line or two to your source document, and your math fonts will be replaced by MathTime Professional fonts. Using LaTeX in the Community, the AoPSWiki, or the Classroom. There are two linear formats for math that Word supports:. Use \begin{document}to start contents and \end{document}to end the document. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. The LaTeX kernel defines several math alphabets in fontmath. Some of them are old type one fonts, others are available as both type one to work with all major TeX engines, or opentype math font to work with LuaTeX and XeTeX using Unicode-math. Another aspect of this dependency is fonts: the serif font used for rendering formulae is browser-dependent and it may be missing some important glyphs. machine learning. Changing Fonts in Mathematics Mode (The following applies to LaTeX2e, a recent version of LaTeX. Installing the Package Content Installing on Unix like Systems Dblatex Packages Dependencies Installation Installing on Windows Dependencies Installation 3. Beautiful Math with LaTeX LaTeX is a powerful markup language for writing complex mathematical equations, formulas, and more. Hartke's »A Survey of Free Math Fonts for T e X and L a T e X« and Walter Schmidt's »Mathematikschriften für L a T e X« (in german). Hartke] Article by Stephen Hartke from Urbana, IL, written in 2006. Preparing Your Ph. How can I change the math font style in matplotlib when using latex. Because MathJax is meant only for math display, whereas LaTeX is a document layout language, MathJax only supports the subset of LaTeX used to describe mathematical notation. NOTE: This page is for jsMath version 2. I'd like to use Microsoft Office to include an equation with a math script font. LaTeX is a high-quality typesetting system; it includes features designed for the production of technical and scientific documentation. tex) is NOTa good model to build a math thesis from. The number of mathematics majors who graduated in 2014-15 was 48; 28 mathematics minors also graduated. Teacher created and classroom approved. The goal of this article is to list all of the free math fonts and to provide examples. To change fonts without using the mouse, use Ctrl-Shift-f and arrow keys. This is my handwriting font for mathematics. Johns Hopkins University, Whiting School of Engineering 3400 North Charles Street, Baltimore, MD 21218-2608 410-516-7210 [email protected] TeX (LaTeX math mode) symbols in legends and Learn more about figure, deep learning vs. You will need to define your fonts at the beginning of any LaTeX document. These can be included in a LaTeX document using the \mathbb{ [letter] } tag from within the math environment. Now if you need to add normal text into a formula or even write a formula using words, you can do this with the text-command inside the math-environment:. This article shows several fonts for use in math mode. I wrote an article about font sizes in LaTeX, which has become a hugely popular post. Latex is a stable dispersion of polymer microparticles in an aqueous medium. Math Equation Converter "LaTex Math, MathType, Mathmagic". For example, you can create APL (programming language) keyboard layout, or for LaTeX/XeTeX, or simply as a system to type math symbols in plain text. 2 The font de nition les LATEX2" knows now new math alphabets called bbm, bbmss and bbmtt. In the above case, they will still be typeset using the CM math fonts, which do not blend well with Times. Just add a line or two to your source document, and your math fonts will be replaced by MathTime Professional fonts. gov to your contacts/address book, graphs that you send yourself through this system will not be blocked or filtered. Depending on your preferred input format, you can create equations in Word in either one of UnicodeMath or LaTeX formats by selecting the format from the Equations tab. 07 A class for typesetting pres. Google Fonts Delivered Free. ) See also \boldsymbol. You can set particular terms in a bold face, and for chemical formulae, you can force the use of an upright font. No extra packages are required to use these symbols. It is intended to cover the majority of users needs, rather than aiming for complete coverage. You set them by using an equals sign: \variable=distance. A long tabbing environment can be split across pages. Since today the latex font on my tablet looks different than yesterday. File -> Export -> Latex will export your lyx file to. Torrents Open Registrations Checker is a simple application that verifies the Torrent/Tracker sites looking for open or closed registrations. LaTeX font size. In LaTex I would use, for example, the command $\mathscr{T}$. PSNFSS provides the default Type 1 fonts listed, excluding Computer Modern (CM), Utopia, Fourier, and Euler. Definitions ewcommand ewenvironment ewtheorem ewfont. Many classes of sets are denoted using doublestruck characters. (See also math fonts and styles. The best website for free high-quality Calibri Math fonts, with 21 free Calibri Math fonts for immediate download, and 55 professional Calibri Math fonts for the best price on the Web. Another aspect of this dependency is fonts: the serif font used for rendering formulae is browser-dependent and it may be missing some important glyphs. Change Font Math Latex Table. Disclaimer: We are checking periodically that all the fonts which can be downloaded from FontPalace. Switch to a large Unicode font like Arial Unicode MS then scroll down to the appropriate script block. 1 Font Sizes This subsection is about font sizes. The sybmol can be compressed to fit on one line (useful for small equations displayed within a text block), or enlarged to make it more readable. Cross References \label \pageref \ref. 07 A class for typesetting pres. Mathematical Bold Fraktur. What I had to do was to right click on the legend in the figure window and then changed the 'interpreter' from 'tex' to 'latex'. This component is what is usually referred to as AMSFonts. No extra packages are required to use these symbols. Blue, white and black are only secondary colours in that context, used for guiding on road surfaces, and for indicating rest areas, etc. , you don't have to pay for it). Do you spend a lot of time typing equations in LaTeX? Try Mathpix Snip for iOS, Android, macOS, Windows or Linux and start converting images to LaTeX instantly!. Each paragraph 1 showcases a different font family and provides some background and usage instructions. Call it $\boldsymbol\beta$. Click to copy — press down alt for multiple Clear As HTML. sty, and a set of virtual math fonts that combine glyphs from Pazo Math, Computer Modern, and Palatino to form complete math fonts for use with TeX/LaTeX are part of the PSNFSS bundle version 9. fd) are needed to inform LATEX2" about the new fonts. We earnestly solicit information from all schools who participate in the development of research level mathematics and from all individuals who may know desired information. The \boldmath declaration switches to a bold math italic font; this causes letters, numbers, and most symbols used in math mode to be set in a bold type. Common documentclass options 10pt/11pt/12pt Font size. Articles must be of interest for a large audience in operations research, written in clear and correct English, and typeset in LaTeX. This post describes how to use LaTeX in an Inkscape drawing, which is probably more useful. I'm confused ! Because I don't use mentiod font in my document !! anyway, I installed the persian-modern fonts but my problem didn't solve ! here is complete output of xelatex :. Preparing Your Ph. twoside Set margins for two-sided. txt) or read online for free. If you use LaTeX, then you would want a tool which converts LaTeX to MathML such as MathType or other online conversion utility. It is here where GUST's e-foundry guys, Bogusław “Jacko” Jackowski, Janusz M. Math fonts in L A T E X have the same five attributes as text fonts: encoding, family, series, shape and size. TIP: If you add [email protected] There are four styles used in typesetting math formulas which affect the size and certain formatting parameters (notably the placement of sub and superscripts on variable size symbols). After switching the math mode font I have faced issues with Greek symbols which had to be typeset in the main (non-math) text. List of LaTeX mathematical symbols. How to specify font size less than 10pt; How can I change the font size in math equations? Styling my section/chapter; Wuerzburg Beamer Theme; LaTeX Beamer Theme Shark; Nice overlay effect in LaTeX Beamer Theme Shark. An example I encountered was reusing LaTeX code from an article in a LaTeX Beamer Class presentation. Offered for use in print, these fonts are delivered using SkyFont’s patent-pending font delivery technology and can be used anywhere. For instance, it's customary to represent real numbers with a blackboard bold font, or topological spaces with calligraphic font. But it isn't. After defining them, you'll only need to use font commands to change the font, for instance to bold or italicize a word or words. Latex is a stable dispersion of polymer microparticles in an aqueous medium. otf-stix AUR - A standalone, more recent version of STIX; otf-latin-modern, otf-latinmodern-math - Improved version of Computer Modern fonts as used in LaTeX. 1 Fonts This section is about fonts. This means, for example, that you cannot put one symbol over another. The Arial Unicode font includes glyphs with a wide variety of uses, including simple math expressions incorporating things like radicals, integrals, and summations. It takes less than a minute to create your account. In the example above the styles remark and definition are used. formulas, graphs). In LaTeX2e , very few fonts are built into the format, and there are commands to load new text and math fonts. Ogni markup matematico deve rientrare all'interno dei due tag … Le interruzioni fisiche di linea all'interno di questi tag non vengono tradotte. In LaTeX, of course, all this stuff is automated: there is a scheme that, for each (text) font size, determines what maths font sizes are to be used. Use Word-to-LaTeX tool to convert any Microsoft Word document to LaTeX, TeX, or clean XML. Hartke's »A Survey of Free Math Fonts for T e X and L a T e X« and Walter Schmidt's »Mathematikschriften für L a T e X« (in german). How to make math font huge; How can I temporarily change the default font size Change font size of \Large etc. Some of them are old type one fonts, others are available as both type one to work with all major TeX engines, or opentype math font to work with LuaTeX and XeTeX using Unicode-math. Although it is quite simple to change the standard text font to a sans serif font, e. michlmayr at gmail. ∞♥ Math Characters / Symbols in HTML ♥∞ For mathematicians, and mathematics educators creating online curricular materials This page explains how to place various mathematical characters on the web via HTML, in lieu of mathemats typesetting software. Here's an image of the left corner of the Word math ribbon showing LaTeX as the current input format. What's the best way to do the equivalent in Microsoft. You can create an account. An online LaTeX editor that's easy to use. Word has a new math ribbon with an explicit LaTeX option as shown in the article Linear format equations using UnicodeMath and LaTeX in Word. The Greek and Cyrillic has been designed under close supervision of an international team of experts, who aimed to set a historical new standard in multi-script type design. Did you know there are so many math fonts?. How to change fonts from Times Roman to Helvetica in Latex:. LaTeX is a programming language that can be used for writing and typesetting documents. LaTeX is a system for preparing documents to be printed or displayed. You can change the cell type to Markdown by using the Cell menu, the toolbar, or the key shortcut m. Package Roman Math Sans serif Typewriter. It provides a unified and user friendly framework for editing structured documents with different types of content: text, mathematics, graphics, interactive content, slides, etc. Hartke] Article by Stephen Hartke from Urbana, IL, written in 2006. However, because of the complexity of latex, lyx can not load very complicated latex files, especially when they use non-standard document style, lots of macros etc. This website provides an overview of basic text formatting commands in LaTeX. more> Accessibility Community. Mathematical Symbols LaTeX-math-mode. more> The Latest. AUCTeX has a built-in way to enter math symbols quickly: see the Entering Mathematics section of the manual. We have about 98 undergraduate mathematics majors and 41 minors. I prefer a math font that is heavier, especially for use on a poster. Renders in the predefined maths bold font. LATEX Mathematical Symbols The more unusual symbols are not defined in base LATEX (NFSS) and require \usepackage{amssymb} 1 Greek and Hebrew letters α \alpha κ \kappa ψ \psi z \digamma ∆ \Delta Θ \Theta. twocolumn Use two columns. LaTeX Commands. , X e T e X 0. The line \usepackage{tgbonum} establishes the font family T e X Gyre Bonum, whose font package name is tgbonum, as the default font for this document. patible math fonts. It is also possible to load any math font with \setmathfont[version=bold]. sty, and a set of virtual math fonts that combine glyphs from Pazo Math, Computer Modern, and Palatino to form complete math fonts for use with TeX/LaTeX are part of the PSNFSS bundle version 9. 2 The font de nition les LATEX2" knows now new math alphabets called bbm, bbmss and bbmtt. The scheme first checks a set of “known” text sizes, for each of which maths sizes are declared in advance. 41 Free Kids Fonts Most Popular - By Name. The doc file won’t render correctly on your machine, however, unless you actually download all the aforementioned fonts. Open an example in Overleaf Changing default font typeface. Anyway, the DEFAULT FONT is a drop down box and it will have alternates after some improvements are made by the developing team. ) The math environment is for formulas that appear right in the text. Making Portable Document Format (PDF) files from LaTeX source is a little tricky, because the PDF file must incorporate not only the images for any figures, but also the font glyphs (or at least, partial fonts) for anything outside the standard handful of fonts in the basic PostScript set. If you need to install fonts on a system without adminstrator privilege, the easiest option is to use math font the MathML-fonts add-on. The default behaviour is to make the font italic, effectively generating the same output as using \textit{}. The name UnicodeMath seems sufficiently different from unicode-math that there shouldn’t be any confusion between the two. Every font is free to download, and 5 are 100% free for commercial-use!. We’ve also listed Symbol although this font is only installed by MathType on Mac (MathType for Windows uses the Symbol font built into Windows). If you want to get the Computer Modern look in MS Word, you have to use a font called Latin Modern. The package kmath uses txfonts for math symbols and uppercase Greek letters. They can be found on the page with Fonts with math support. For instance, it's customary to represent real numbers with a blackboard bold font, or topological spaces with calligraphic font. …Cambria Math is the only supported font that can be used in Equation Editor in Word 2013 for the time being. sty [1996/05/31 STEP AP package][CS] iso10303 ap version 1. Johns Hopkins University, Whiting School of Engineering 3400 North Charles Street, Baltimore, MD 21218-2608 410-516-7210 [email protected] Depending on your preferred input format, you can create equations in Word in either one of UnicodeMath or LaTeX formats by selecting the format from the Equations tab. A Beginner’s Guide to LATEX David Xiao [email protected] If the fonts you are missing don't begin with "jsMath-", you should use the page for version 2. it into a latex le. That does not mean that only these three sizes can be used, it is only the size of the normalsize font. For some reason, the ieeconf class uses Times font for regular text and Computer modern roman for math text. The jsMath package is designed to work best if you have installed the TeX font set. Hints: No $$\texttt{\}$$ signs needed; All formulas are rendered in display style. Other than that, thank you. If at the time will discount more Savings So you already decide you want have Change Font Math Latex Table for your, but you don't know where to get the best price for this Change Font Math Latex Table. You don't have to pay for using LaTeX, i. The Comprehensive LaTeX Symbol List. Using R Markdown for Class Reports - stat. Produce your own math paper, full of research-level, professionally formatted nonsense! Just enter your name and those of up to 3 "co-authors". I am using the ieeconf latex class to write a document. Using Bold Greek Letters in LaTeX. 07 A class for typesetting pres. For the complete alphabet, see: Math Font ℤ ℚ ℝ ℂ ⅈ ℑ ℜ ℭ ℵ; Greek α β γ; Look-Alike Math Characters. What I had to do was to right click on the legend in the figure window and then changed the 'interpreter' from 'tex' to 'latex'. It now can do PNG and SVG images exports. By default, Latex will print text within formulas in italics, omitting white spaces. If you really want to know, the default LaTeX font is called Computer Modern, and it is a member of the Didone family of fonts, which IMO are generally suited to titling or posters but not body text (which is why you've probably never read a book that was set in one). Changing Fonts in Mathematics Mode (The following applies to LaTeX2e, a recent version of LaTeX. To learn how this works, I suggest choosing an example from the the "LaTeX Examples" drop-down list at the lower left. For example, you can create APL (programming language) keyboard layout, or for LaTeX/XeTeX, or simply as a system to type math symbols in plain text. Nine tabs offer quick access to frequently used features, including those we add ourselves. If the fonts you are missing don't begin with "jsMath-", you should use the page for version 2. A few sections in User's Guide explain how to set up your LaTeX projects in order to take full advantage of WinEdt's capabilities when it comes to navigating in large projects or collecting data for purpose of referencing and citations. You will not easily find it on MS Word. Mathematical Equation Editor and Conversion. If you are looking for bold math or something like it, have a look at the AMS fonts. For a complete list of available font sizes see the reference guide. Find the best 13 free fonts in the Math style. I really wonder if it is possible to change WinEdt's font and size in the compiler window? Follow Math Help Forum on Facebook and Google+ Sep 7th 2008, 05:33 AM #4. Full Text Search. To use LaTeX markup, set the Interpreter property for the Text object to 'latex'. Note that you only need to download ONE version of the ttf file for each font. These LaTeX's symbols are grouped together more or less according to function. TeX documents are written and programmed using an unusual macro language. Latex is a stable dispersion of polymer microparticles in an aqueous medium. 2 sscript-features Fontfeaturesfornestedsub-andsuper-scripts section§4. For when it’s too beautifully typeset to be wrong: proof by LaTeΧ. ∞♥ Math Characters / Symbols in HTML ♥∞ For mathematicians, and mathematics educators creating online curricular materials This page explains how to place various mathematical characters on the web via HTML, in lieu of mathemats typesetting software. Fonts that specifically use the Computer Modern math encodings should use 7m', 7v', and 7y'. This is because Latex does not use computer modern for Greek letters, but rather a font which (in Ubuntu, at least) is called cmmi10 font, as I found by looking at the PDF. Vincent, who is a die-hard fan of the command line, recommends using the command-line editor nano. Il importe donc de ne retenir, parmi les solutions que l'on peut trouver sur le Net, que celles qui utilisent le système NFSS. Macro language. Although it is quite simple to change the standard text font to a sans serif font, e. Variable symbol commands. Free math lessons and math homework help from basic math to algebra, geometry and beyond. Font Format and Supported Typesetting Systems. An \oval command in a picture specified an oval that is smaller than latex can make it. The line \usepackage{tgbonum} establishes the font family T e X Gyre Bonum, whose font package name is tgbonum, as the default font for this document. It was created specifically by Donald Knuth, the creator of the TeX typesetting system, for LaTeX. the margins, the font, the spacing, the document type, supplementary packages, etc. So maths has to be set using a standard LaTeX font. The font can also be changed for a specific element in the document. On my sites I have math only on a very few pages, and MathJax-LaTeX enables me to enable mathjax only on those pages by just adding the [mathjax] shortcode somewhere. Did you know there are so many math fonts?. Mission Statement. Same with the Rational, Real, and Natural domain symbols (although their values won't be 2124). Click OK and print. They cannot be used directly in the source. You can designate the text color on your web page using the 'color' attribute in the HTML font element. tex-gyre-math - Maths fonts to match tex-gyre text fonts T e X-Gyre-Math is a collection of maths fonts to match the text fonts of the T e X-Gyre collection. Beautiful Math with LaTeX LaTeX is a powerful markup language for writing complex mathematical equations, formulas, and more. The Asana-Math font is an OpenType font that includes almost all mathematical Unicode symbols and it can be used to typeset mathematical text with any software that can understand the MATH OpenType table (e. It is found in nature, but synthetic latexes can be made by polymerizing a monomer such as styrene that has been emulsified with surfactants. Font Sizes \tiny \scriptsize \footnotesize \small \normalsize \large \Large \LARGE \huge \Huge All of these fonts are listed from smallest to largest. Maths fonts to match T e X. If you would like to produce full documents of your own LaTeX, please follow the directions below. However, it cannot apply for Powerpoint because the Equation Option dialog in Powerpoint differs from the one in Word. For math fonts, the most important aspect is it's spacing, optical sizing, and accent placement, and weights etc. Maths fonts to match T e X Gyre Schola. A Survey Of Free Math Fonts For LaTeX TUG Feb 3, 2006 - Computer Modern, CM Bright, Concrete and Euler, Concrete Math,. You might have read the 36 methods of mathematical proof, and I request an addition. The standard type sizes in Latex are: tiny, scriptsize, footnotesize, small, normalsize, large, Large, LARGE, huge, Huge To change the size within the math environment, try the following $$\mbox{\Huge 3x+3=\mu }$$ You can define a short alias command for creating bold math notation in the. This is a partial list of commercial vendors and other non-free sources of TeX and TeX-related software. In Word 2016 (Windows, Office 365 users only), you can see some extras on the Equation Editor ribbon, especially the LaTeX. tex-gyre-math - Maths fonts to match tex-gyre text fonts T e X-Gyre-Math is a collection of maths fonts to match the text fonts of the T e X-Gyre collection. Common documentclass options 10pt/11pt/12pt Font size. 𝕬 𝕭 𝕮 𝕯 𝕰 𝕱 𝕲 𝕳 𝕴 𝕵 𝕶 𝕷 𝕸 𝕹 𝕺 𝕻 𝕼 𝕽 𝕾 𝕿 𝖀 𝖁 𝖂 𝖃 𝖄 𝖅. Torrents Open Registrations Checker is a simple application that verifies the Torrent/Tracker sites looking for open or closed registrations. Here’s an image of the left corner of the Word math ribbon showing LaTeX as the current input format. I haven't seen it yet in CTAN, though. texin section 4. LaTeX symbols have either names (denoted by backslash) or special characters. Latex is a stable dispersion of polymer microparticles in an aqueous medium. Math Boxes and glue Boxes Inviolable space required by some typsetting element Could be one letter Has aselineb , height , width , and depth LaTeX Spacing Tricks. Franck indicates here that the definite way is to download those fonts from there and install. AMS-LaTeX If you're typing math, you should check out (and use) AMS-LaTeX. 0 and above.
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# How do you solve the area of a trapezoid using diagonals
The height of a trapezoid is $10$ cm. The lengths of the two diagonals of the trapezoid are $30$ cm and $50$ cm. Calculate the area of the trapezoid.
On the homework I solved this using $${D_1D_2\over 2}$$ and my teacher marked me wrong. So I don't know what I did wrong. Please help. I know I can only use the formula if the diagonals are $90$ degrees. But how do I check that ?
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that is the formula of the area of a rhombus. – lab bhattacharjee Dec 20 '12 at 5:02
To help you think about the solution, draw two parallel lines 10 cm apart. The bottom will contain the B1 and the top will contain the B2. Draw the two diagonals at the proper lengths from the base to the top line making sure they cross. Now imagine that you slide one of the diagonals along the bottom and top lines. Notice that (B2+B2)/2 does not change (i.e one stretches and one shrinks). So no matter where the diagonals cross you have the same area. Now slide them apart until the intersection point reaches the top line (i.e. B2 = 0). Now you have triangle with the same area as the trapezoid. You know the height and two sides of the triangle, a little geometry and you can compute the base and your're there.
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The area will be $\frac12\cdot 10\cdot (y+x+y+z)=5(x+2y+z)$
Now, $(y+z)^2+10^2=50^2$ and $(x+y)^2+10^2=30^2$
$(y+z)=\sqrt{50^2-10^2} CM=20\sqrt6 CM$
$(x+y)=\sqrt{30^2-10^2} CM=20\sqrt2 CM$
SO, the area will be $5(20\sqrt2(\sqrt3+1)) CM^2$
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Gluing two copies of the trapezoid together with one rotated by 180 degrees can yield a parallelogram. One diagonal from each trapezoid cuts the parallelogram into two triangles, each of which has an altitude of 10 and sides adjacent to that altitude's vertex of lengths 30 and 50.
So the area is the same as that of a triangle with two sides of 30 and 50, and an altitude between them of 10. Is this enough of a help?
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Trying to understand what you said... Is the answer 193.1 cm? – NGPP1 Dec 20 '12 at 0:35
This is the right answer, but I don't think your gluing construction works. – Peter Shor Dec 20 '12 at 0:36
I am still confused – NGPP1 Dec 20 '12 at 0:42
@PeterShor, in my method, the area is around $386.37 cm^2$ – lab bhattacharjee Dec 20 '12 at 5:03
@PeterShor I have added imagery that demonstrates the gluing construction works. I leave the labeling of edges and confirmation to you. The trapezoid's area is the same as that of the triangle at the end of the process: a triangle with altitude 10 and two surrounding sides of length 30 and 50. – alex.jordan Dec 20 '12 at 10:05
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# How to represent point-at-infinity in affine coordinate
In projective coordinates point-at-infinity can be identified with z=0. How to identify the point-at-infinity in affine coordinate.
Whether x=0 and y=0 can be considered as point-at-infinity in affine coordinate?
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Does crypto.stackexchange.com/questions/6156/… help, by any chance? – Thomas Feb 25 '13 at 5:58
@Thomas: I went through the link you posted. My question is not related to encoding of the point-at-infinity. In our implementation after EC operation such as EC point addition we return the point "x=0, y=0" when we encounter point-at-infinity. So I wanted to check is it safe in doing this? Is there a remote possibility of obtaining a valid point with x=0,y=0 when working with standard EC curves? – Andy Feb 26 '13 at 4:15
Well, clearly (0,0) is a valid point for the curve y^2=x^3+x which is in the standard reduced Weierstrass form. If you want a simple encoding that can be unambiguous, why not use (p,0) where p is the cardinality of the field of definition ? – minar Jul 3 '13 at 22:27
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# Oxide content description
From time to time, I come by to see what people do with my mhchem package. Here, I came across a notation that I did not know yet: $\ce{K2O-nSiO2-xH2O}$ and $\ce{K2O - 3.2SiO2 - 2.7H2O}$
What is this? Is this really a bond? Is this a short form for writing $\ce{K2O-Si_nO_{2n}-H_{2x}O_x}$ and $\ce{K2O - Si_{3.2}O_{6.4} - H_{5.4}O_{2.7}}$? Is this an established notation that mhchem should support? (It doesn't yet with the numbers.) Do you have authoritative References?
• My gut feeling is that they should be hyphens. – Jan Oct 2 '17 at 11:56
• Dots, if anything. $\ce{K2O\cdot nSiO2}$... – Ivan Neretin Oct 2 '17 at 12:26
• I don't think this is standard notation, and the linked question suggests that it is rather a recipe or composition of a mixture of salts rather than an actual compound formula. – Martin - マーチン Oct 2 '17 at 12:28
• Wait, you're the author of mhchem?! – Zhe Oct 2 '17 at 12:59
• @IvanNeretin Dots make even more sense. I had thought the manufacturer had at least gotten the type of symbols right so my brain was stuck in the distinction between different types of horizontal lines ^^' – Jan Oct 3 '17 at 5:57
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# teaching machines
## CS 240: Lecture 38 – Graph Algorithms
November 29, 2021 by . Filed under cs3, fall-2021, lectures.
Dear students:
I don’t know about you, but my mind has been elsewhere. Our routine striving was cut into by a break. I’m glad it did. Sometimes I think that I’ll only have peace when I get all the things done. What a lie that is. I will never be done. And even if something I’m working on gets finished, I won’t have peace afterward, not for long anyway. It’s better that breaks just interrupt us without waiting for a convenient time.
We pick up our conversation of algorithms and data structures with a couple of tricks for finding meaningful paths through graphs. First we’ll find a path that navigates through a prerequisite chain. Second we’ll find the shortest path from a vertex to another. These problems are pretty easy for humans to solve visually when the number of vertices is small. Don’t be deceived. As with every other topic in this class, we must assume that the size of our data structure will exceed our capacity to deal with it. Yet when we introduce the topic, we use small examples.
### Topological Sort
Imagine a series of tasks in a multi-task process. Some tasks must be completed before others. For example, maybe you’re planning a space flight. You’ve got to get some new o-rings before you assemble the solid rocket booster before you install the booster. Other tasks can be scheduled more freely, like celebrating a successful launch with a glass of Blue Powerade. They can happen any time after earlier stages. You can find a path through this process by making each stage a vertex. If a stage is a prerequisite for another, then there’s a directed and unweighted edge from the prerequisite to its successor. Then you run a topological sort on the graph.
The Greek word topos means place. The topological sort orders the vertices by their place in the prerequisite chain, not by any numeric value. The algorithm can be implemented a couple of different ways. We can run a depth-first traversal, chasing out the longest prerequisite chain in the graph and adding its vertices in backward order and branching recursively through the neighbors. A pseudocode implementation might look like this:
function topologicalSort(graph g)
for each vertex v
mark v as unvisited
order = []
for each vertex v
if v is unvisited
depthTraverse(g, v, order)
function depthTraverse(graph g, vertex v, list order)
mark v as visited
for each neighbor n
if n is unvisited
depthTraverse(g, n)
prepend v to order
Alternatively, if we wish to avoid the recursion, we can implement this with a breadth-first traversal. This implementation works by keeping a count of how many incoming edges each vertex has from unvisited vertices. When that number goes to 0, a vertex’s prerequisites have been visited and thus it is safe to visit the vertex. Here’s how it might be implemented in pseudocode:
function topologicalSort(graph g)
create queue
order = []
for each vertex v
inCounts[v] = 0
for each vertex v
for each neighbor n
increment inCounts[n]
for each vertex v
if inCounts[v] is 0
enqueue v
while queue isn't empty
v = dequeue
append v to order
for each neighbor n
decrement inCounts[n]
if inCounts[n] is 0
enqueue n
There are more legal topological sorts than these deterministic algorithms will generate. Consider this graph:
How many possible topological sorts are there for this graph? We’ll answer this as a peer instruction exercise.
### Shortest Paths
The second algorithm we’ll look at today finds the quickest route from a starting vertex to all the other vertices in the graph. This algorithm was invented by Edsger Dijkstra, an opinionated computer scientist from the Netherlands. It works by maintaining a record of how distant each vertex is. Initially all vertices are infinitely far away, except for the starting vertex, which has a distance of 0. It advances to the next nearest vertex. If that vertex makes reaching other vertices cheaper, then those vertices’ distances are lowered. Then it advances again and again, until all vertices have been visited.
To actually trace out a path, the algorithm must keep track of which vertex provided the cheapest route. We walk this list of predecessors backward to generate the complete path.
function shortestPaths(graph g, vertex from, int[] distances, vertex[] predecessors)
for each vertex v
distances[v] = infinity
predecessors[v] = null
mark v as unvisited
distances[from] = 0
repeat g.vertices.size
v = find closest unvisited vertex
mark v as visited
if distances[v] is infinity
return as there's no way to reach any more vertices
for each neighbor n
distanceViaV = distances[v] + g.weight(v, n)
if distances[n] > distanceViaV
distances[n] = distanceViaV
predecessors[n] = v
To find the shortest path to a particular destination, we trace backward through the predecessors list, starting at the at destination and stopping at from.
### Exercises
For the remainder of our time, we will work through an exercise on Dijkstra’s algorithm.
### TODO
You have some work to do before the next class:
• Keep working on PA4. It is due on Wednesday.
• Complete the quizzes by the end of today.
• Submit the lab by the end of today.
• There are no more OpenDSA readings.
See you next time.
Sincerely,
P.S. It’s time for a haiku!
Course N won’t make sense
Not until course N + 1
Thus post-requisites
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# Question 2 Dragsters can achieve average accelerations of 26.0 m/s2 Suppose such a dragster accelerates from rest at this rate
###### Question:
Question 2 Dragsters can achieve average accelerations of 26.0 m/s2 Suppose such a dragster accelerates from rest at this rate for 5 5. How far in meters will the dragster travel? B. Dragsters can achieve average accelerations of 26.0 m/s2. Suppose such a dragster accelerates from rest at this rate for 4.39 s. How far does it travel in this time? Suppose a car merges into freeway traffic on a 313-m-long ramp. If its initial velocity is 10.0 m/s and it accelerates at 2.00 m/s2, how long does it take to travel the 313 m up the ramp? (Such information might be useful to a traffic engineer:)
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# Generate random number from $P(x)=Ax^{-3/2}\exp(-B/x)$?
I wish to numerically generate a random number from the distribution $$P(x) = Ax^{-3/2}\exp(-B/x),\qquad A,B,x>0$$
What would be the easiest way of achieving this?
Inverse transform sampling does not seem appropriate since the above $P(x)$ does not have an analytical cumulative distribution function.
Rejection sampling requires a second distribution to draw samples from, but I don't know what known distribution would be efficient for this purpose.
If $X$ is distributed according to your density, then $\frac{2B}{X}$ has density proportional to $x^{-\tfrac{1}{2}} e^{-x/2}$. This is the density of a Chi-squared distribution with one degree of freedom, i.e. $X \stackrel{d}{=} \frac{2B}{N^2}$, where $N$ is a standard normal random variable.
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## Abstract
Microdeletions of Yq are associated with azoospermia and severe oligozoospermia. In general, men with deletions are infertile and therefore deletions are not transmitted to sons unless in-vitro fertilization (IVF) and intracytoplasmic sperm injection (ICSI) are performed. We report an unusual family characterized by multiple members with infertility and Yq microdeletion. Complete reproductive history, semen analyses and blood samples were elicited from relevant family members. DNA preparation and quantification were performed using commercial kits. A total of 27 pairs of sequence tagged sites based primer sets specific for the Y microdeletion region loci were used for screening. Southern blots using deleted in azoospermia (DAZ) and ribosomal binding motif (RBM) cDNAs were then analysed for confirmation. The proband, his three brothers and father were all found to be deleted for DAZ but not RBM. At the time of analysis, the proband's father was azoospermic whereas his four sons were either severely oligozoospermic or azoospermic. Unlike their father, the four sons are infertile and have no offspring, except for one of them who achieved a daughter only after IVF/ICSI treatment for infertility. Microdeletions of Yq involving the DAZ gene are associated with a variable phenotypic expression that can include evidently normal fertility.
## Introduction
Infertility occurs in ~14% of couples (Mosher, 1985) and abnormalities in the male partner are estimated to be present in up to half of the cases (Swerdloff et al., 1985). Efforts to evaluate the causes of azoospermia have shown that after exclusion of traditionally recognizable causes (i.e. abnormal karyotype, obstruction, varicocoele, hormonal defect, etc.), most cases (50–75%) are unexplained and are termed idiopathic (Pryor et al., 1997). Recently, it has been reported that up to 30% of men with idiopathic' azoospermia have microdeletions of the Y chromosome (Henegariu et al., 1993; Ma et al., 1993; Nagafuchi et al., 1993; Kobayashi et al., 1994; Najmabadi et al., 1996; Reijo et al., 1996; Vogt et al., 1996; Pryor et al., 1997). Exactly how and whether or not these microdeletions cause azoo/oligozoospermia is the subject of both intense investigation and debate.
Essential to the argument that Y microdeletions cause infertility is the observation that fertile men rarely manifest Y microdeletions. Microdeletions in four out of 200 fertile men studied have been reported (Pryor et al., 1997). However, the deletions in these men were very small and most likely represented insignificant polymorphism. Relatively large deletions of the kind associated with male infertility have not been reported in men with normal fertility. Although it is generally assumed that these deletions arise de novo and that father to son transmission of Y microdeletion would not be expected, a few rare instances of father to one son transmission of Y chromosome microdeletion have been reported (Kobayashi et al., 1994; Stuppia et al., 1996; Vogt et al., 1996; Pryor et al., 1997). However, vertical transmission of a microdeletion involving the deleted in azoospermia (DAZ) locus from father to one son has been reported in only three cases (Kobayashi et al., 1994; Vogt et al., 1996; Pryor et al., 1997). We now describe a four-generation family in which an azoospermic father and his four infertile sons all share an apparently identical microdeletion that includes the DAZ locus. This family represents the first and only report of spontaneous vertical transmission of DAZ deletion to multiple offspring. It provides evidence that a single Yq microdeletion can result in varying phenotypic expression in different individuals. It is clinically significant, in that the presence of a microdeletion is not an absolute marker for infertility and can be associated with apparently normal fertility.
## Materials and methods
Screening for Yq microdeletion was performed on a routine basis for male infertility using a protocol reviewed and approved by the Institutional Review Board of College of Physicians & Surgeons, Columbia University. Samples were taken from patients after informed consent.
### Semen analysis
Results were analysed using WHO criteria with a Nikon phase contrast microscope.
### Serum hormone concentrations
Follicle stimulating hormone (FSH), luteinizing hormone (LH), and testosterone were measured by solid-phase, two site chemiluminescent enzyme immunometric assay (Immulite; Diagnostic Products Corporation, Los Angeles, CA, USA). Normal ranges for men are FSH <10 mIU/ml; LH <10 mIU/ml; and testosterone 270–1070 ng/dl.
### Genomic DNA
Extraction of genomic DNA from whole blood was performed by lysis of red blood cells, followed by lysis of white blood cells and their nuclei. Cellular proteins were removed by salt precipitation, and genomic DNA was precipitated with isopropanol using Puregene DNA extraction kit (Gentra Systems, Inc. Minneapolis, MN, USA; catalogue no. D-5004).
### Polymerase chain reaction (PCR)
Primers were produced as dried oligonucleotides on an automated DNA synthesizer (Perkin-Elmer Applied Biosystems Inc., Foster City, CA, USA). A total of 27 Y chromosome specific sequence tagged sites (STS) (Figure 1) were selected from an STS map (Vollrath et al., 1992). They include the three proposed spermatogenesis loci AZFa, AZFb, and AZFc (as per Vogt et al., 1996) spanning Yq intervals 5, 6 and 7. As a rapid screening protocol, a PCR multiplex system composed of two to six different primer pairs was used in a total of six multiplexed reactions (Table I). With each PCR run, a female control and a normal male control were included. All PCR reactions were run in polycarbonate (Techne®) plates in an MJ Research® machine. The PCR conditions were essentially as previously described (Henegariu et al., 1993). Briefly, in a 14 μl total volume reaction, 50 ng of genomic DNA was used as template, 1 μl of primer standard solution (mix I or II or III or IV or V or VI consisting of 10 pmol per primer), 12 μl of PCR cold mix' (1.5 mmol/l MgCl2, 0.2 mmol/l of each dNTP, 5% DMSO, 1× Taq polymerase reaction buffer without Mg2+), 1.25 IU Taq DNA polymerase (Promega) and 1 drop of oil. The complete mixes were placed directly in a thermocycler preheated to 94°C. Cycling conditions for 27 cycles were: 94°C, 30 s (melting); 55°C, 45 s (annealing); and 72°C, 60 s (extension). The final extension time was 5 min. The PCR reaction products were then separated on 3% agarose gels (Bio-Rad, ultra-pure grade) by electrophoresis in TBE buffer. PCR products were stained with ethidium bromide and visualized by exposure to ultraviolet light. STS showing no amplification in multiplex reactions were confirmed by single reaction PCR with appropriate positive and negative controls. An STS was considered to be absent after three amplification failures.
### Southern hybridization
Southern blotting was performed according to established protocol (Sambrook et al., 1989). Briefly, 5 μg genomic DNA was digested with HindIII or TaqI, run on a 0.7% agarose gel in standard TBE buffer, transferred to a nylon membrane, and hybridized with 32P-labelled probes. The DAZ probe was the purified insert of a plasmid (pDP1577) containing the full length cDNA (Reijo et al., 1995). Similarly, the RBM probe was the plasmid insert of an RBM cDNA clone (MK5) (Ma et al., 1993).
### Paternity determination
Paternity of all four sons was confirmed by showing the expected segregation of four highly polymorphic autosomal markers (Weber and May, 1989). These were D21S156, D21S270, D13S132, and D13S159 with heterozygosities of 0.83, 0.86, 0.84 and 0.90 respectively.
### Fluorescent in-situ hybridization (FISH)
FISH for DAZ was performed with Cosmid 63C9 (Saxena et al., 1996), using established methods (Yu et al., 1996).
## Results
The finding that individual II-8 in the pedigree was azoospermic but did not have a microdeletion was a surprise. The DAZ locus in this individual was further tested by Southern analysis using the DAZ cDNA and a different restriction enzyme (TaqI). It failed to show any abnormality of the DAZ locus. In addition, FISH with the DAZ Cosmid 63C9 showed normal intensity (data not shown).
The proband (III-8) and his older brother (III-6) were seeking infertility treatment. After extensive counselling, they opted for in-vitro fertilization (IVF) and intracytoplasmic sperm injection (ICSI). The proband (III-8) and his wife (III-9) underwent two IVF-ICSI cycles that failed to produce a pregnancy secondary to poor ovarian response. The proband's brother (III-6) and his wife (III-7) underwent one cycle of controlled ovarian hyperstimulation and nine oocytes were retrieved. Eleven mature spermatozoa were found in three ejaculates on the day of retrieval and used for ICSI. Three oocytes fertilized which subsequently cleaved and were transferred. She delivered a healthy female baby (IV-2).
## Discussion
We report an exceptional family in which an azoospermic father and his four infertile sons share an apparently identical Yq microdeletion involving the DAZ locus. The de-novo mutation that led to the microdeletion of DAZ appears to have originated in the proband's father (II-1). This deletion is expected to cause azoo/oligozoospermia and male infertility, and yet he spontaneously conceived five children and was unaware of any fertility problem. Interestingly, however, all four sons are infertile and are either azoospermic or severely oligozoospermic.
This family raises several issues with regards to the association between Yq microdeletion and infertility. First, it confirms that vertical transmission of Yq microdeletion is possible and can lead to subsequent infertility in the male offspring. Second, it is obvious that the same deletion can result in different phenotypes in different individuals. Although the father (II-1) of the four boys in this family was azoospermic at the time of analysis, he fathered his first child at the age of 25 and his last one at the age of 38 years. Thus, he possessed some degree of fertility over a large span of years. Likewise, microdeletion of the Y chromosome, specifically DAZ, does not necessarily imply a lifelong history of azoospermia nor does it preclude the formation of a large family. His four sons, on the other hand, are infertile and either azoospermic or severely oligozoospermic.
The DAZ gene has been proposed as the azoospermia factor on the Y chromosome. This family shows clearly that while DAZ may have a critical role in spermatogenesis, it is not essential for fertility. Furthermore, total loss of the DAZ gene cluster can be associated with a histological picture of Sertoli cell only' as well as sperm maturation arrest (Foresta et al., 1997; Pryor et al., 1997). Several authors have found a poor correlation between the location of Y microdeletions (including DAZ deletions) with the clinical and histological phenotype of the patients (Reijo et al., 1995, 1996; Vogt et al.; 1996; Silber et al., 1998). The findings in this family agree that such a correlation may turn out to be quite problematic. Testicular biopsy of the proband's brother (III-6) showed a picture of Sertoli cell only' whereas the proband (III-8 with sperm count 0.5×106/ml) clearly would be expected to have some degree of sperm maturation on biopsy. Furthermore, testicular biopsy may not be representative of the entire testicle because there may be geographic heterogeneity for spermatogenesis as in individual III-6 whose ejaculates contained mature spermatozoa.
We can only speculate about the basis for phenotypic differences between family members with the same deletion. It is well known that identical deletions within autosomes may result in different phenotypes (Schinzel, 1994). One can postulate that such differences are consequences of each individual's exposure to his environment or expression of various modifying genes. A fertile father has been described with a microdeletion that widened when transmitted to his infertile son (Stuppia et al., 1996). Although variable extensions at the borders of the deletion may exist between our different family members, these molecular extensions cannot be distinguished by interval mapping. By PCR analysis, the same STSs failed to amplify in our five individuals and Southern hybridization with the DAZ probe confirmed a complete deletion of this gene cluster. Although it is possible that the deletions observed are, in fact, not identical and adjacent areas may contain important genes that modulate the degree of phenotypic expression, these results still indicate a large overlap of deleted Y DNA (including the loss of DAZ gene cluster) in each individual of this unique family.
We were fascinated by the fact that the proband's uncle (II-8) has infertility and azoospermia but no apparent microdeletion by STS testing. Since southern blot analysis using both the RBM and DAZ probes as well as FISH analyses using a DAZ cosmid were all entirely normal, we are forced to conclude he has a different aetiology underlying his infertility. Admittedly, it is possible that he may have a smaller or point mutation/perturbation or proximal/distal rearrangement that is not detectable by current methods. He gave no history of exposure to gonadotoxins or other definable factors that were likely to affect spermatogenesis.
Until recently, Y microdeletion has had little clinical significance, since a man with a deletion will not, in general, reproduce. However, utilizing ICSI and testicular sperm aspiration (TESA), combined with IVF, it is now possible for oligo/azoospermic men with Y microdeletion to achieve pregnancies (Mulhall et al., 1997; Silber et al., 1998, and individual III-6). This has fostered concerns that such pregnancies may produce male offspring with similar microdeletions and subsequent infertility (Reijo et al., 1996; Girardi et al., 1997; Kremer et al., 1997). Indeed, Yq microdeletion can be transmitted to male offspring via ICSI (Kent-First et al., 1996). The family we report suggests that men with Yq microdeletions (such as individual II-1) who achieve pregnancies will transmit the same microdeletion and the risk of infertility to their sons (individuals III-1, III-4, III-6, III-8). Therefore, patients should be offered Y microdeletion screening prior to ICSI and they should be counselled on the certainty of transmitting the Yq microdeletion and possibly infertility to their sons. As more research is focused on genetic aetiologies of male infertility, identification of genes involved in spermatogenesis should provide insight into the pathophysiology of male infertility and a more rational basis for initiating therapy.
Table I.
Multiplex polymerase chain reaction (PCR) scheme used for the 27 STS primer pairs. The primers are ordered by decreasing expected lengths
Multiplex mix Sequence tagged site (STS) Expected PCR product length (bp) Corresponding locus
157 285 DYS240
154 245 DYS238
142 196 DYS230
145 160 DYF51S1
131 143 DYS222
139 120 DYS227
II 134 301 DYS224
136 235 DYS226
129 194 DYS220
132 159 DYS7
152 125 DYS236
III 143 311 DYS231
55 256 DYF67S1
130 173 DYS221
149 132 DYS1 (DAZ
147 100 DYS232
IV 83 275 DYS11
158 231 DYS241
148 202 DYS233
138 170 DYF49S1
153 139 DYS237
164 690 DYF65S1
84 326 DYS273
87 252 DYS275
144 143 DYF50S1
VI 159 550 DYZ2
160 236 DYZ1
Multiplex mix Sequence tagged site (STS) Expected PCR product length (bp) Corresponding locus
157 285 DYS240
154 245 DYS238
142 196 DYS230
145 160 DYF51S1
131 143 DYS222
139 120 DYS227
II 134 301 DYS224
136 235 DYS226
129 194 DYS220
132 159 DYS7
152 125 DYS236
III 143 311 DYS231
55 256 DYF67S1
130 173 DYS221
149 132 DYS1 (DAZ
147 100 DYS232
IV 83 275 DYS11
158 231 DYS241
148 202 DYS233
138 170 DYF49S1
153 139 DYS237
164 690 DYF65S1
84 326 DYS273
87 252 DYS275
144 143 DYF50S1
VI 159 550 DYZ2
160 236 DYZ1
Table II.
Semen analyses and hormone profiles of relevant family members
ID Relationship to proband Age (years) Sperm count (×106/ml) FSH (mIU/ml) LH (mIU/ml) Testosterone (ng/dl)
FSH = follicle stimulating hormone; LH = luteinizing hormone; NA = not analysed.
III-8 Proband 24 0–0.5 3.5 4.5 485
III-6 Brother 33 3 spermatozoa 5.1 2.5 279
III-4 Brother 37 0.1 5.5 1.7 499
III-1 Brother 38 NA 6.3 1.6 414
II-1 Father 63 21.2 3.3 392
II-8 Uncle 44 40.7 8.7 37
Normal values >20 <10.0 <10.0 270–1070
ID Relationship to proband Age (years) Sperm count (×106/ml) FSH (mIU/ml) LH (mIU/ml) Testosterone (ng/dl)
FSH = follicle stimulating hormone; LH = luteinizing hormone; NA = not analysed.
III-8 Proband 24 0–0.5 3.5 4.5 485
III-6 Brother 33 3 spermatozoa 5.1 2.5 279
III-4 Brother 37 0.1 5.5 1.7 499
III-1 Brother 38 NA 6.3 1.6 414
II-1 Father 63 21.2 3.3 392
II-8 Uncle 44 40.7 8.7 37
Normal values >20 <10.0 <10.0 270–1070
Figure 1.
Y chromosome map and microdeletions in subinterval 6D–6F of the Y chromosome long arm in the proband (III-8), his father (II-1) and three brothers (III-1, III-4, III-6). The presence of a sequence tagged site (STS) is indicated by the solid portion of the column. The STS not amplified are marked with asterisks. The approximate boundaries of AZFa, AZFb, and AZFc regions (as per Vogt et al., 1996) are shown.
Figure 1.
Y chromosome map and microdeletions in subinterval 6D–6F of the Y chromosome long arm in the proband (III-8), his father (II-1) and three brothers (III-1, III-4, III-6). The presence of a sequence tagged site (STS) is indicated by the solid portion of the column. The STS not amplified are marked with asterisks. The approximate boundaries of AZFa, AZFb, and AZFc regions (as per Vogt et al., 1996) are shown.
Figure 2.
Pedigree of the four-generation family with results of Yq microdeletion testing. The proband (III-8) is indicated by an arrow. The proband (III-8) is severely oligozoospermic and microdeleted for subinterval 6D-6F of Yq. His father (II-1) was found to be azoospermic and the two brothers (III-4, III-6) were severely oligozoospermic. The third brother (III-1) declined semen testing. The proband, his father and three brothers were all found to have an apparently identical microdeletion including DAZ. The proband's uncle (II-8) was found to have azoospermia but no Yq microdeletion was detected.
Figure 2.
Pedigree of the four-generation family with results of Yq microdeletion testing. The proband (III-8) is indicated by an arrow. The proband (III-8) is severely oligozoospermic and microdeleted for subinterval 6D-6F of Yq. His father (II-1) was found to be azoospermic and the two brothers (III-4, III-6) were severely oligozoospermic. The third brother (III-1) declined semen testing. The proband, his father and three brothers were all found to have an apparently identical microdeletion including DAZ. The proband's uncle (II-8) was found to have azoospermia but no Yq microdeletion was detected.
Figure 3.
Southern blot with DAZ probe. The entire DAZ locus in individuals II-1, III-8, and III-6 is absent, whereas it appears to be present and normal in the control male and individuals I-1, II-8 and IV-1.
Figure 3.
Southern blot with DAZ probe. The entire DAZ locus in individuals II-1, III-8, and III-6 is absent, whereas it appears to be present and normal in the control male and individuals I-1, II-8 and IV-1.
Figure 4.
(a) Metaphase from individual I-1 (control) after FISH using probe DYZ3 (ONCOR) to identify the Y centromeric DNA (green) and probe Cosmid 63C9 to identify the DAZ-containing chromosome region (red). Both signals are seen on the Y chromosome. (b) Metaphase from individual II-1 using the same probes. Only the green signal is seen, identifying the Y chromosome, but no signal for DAZ region is present.
Figure 4.
(a) Metaphase from individual I-1 (control) after FISH using probe DYZ3 (ONCOR) to identify the Y centromeric DNA (green) and probe Cosmid 63C9 to identify the DAZ-containing chromosome region (red). Both signals are seen on the Y chromosome. (b) Metaphase from individual II-1 using the same probes. Only the green signal is seen, identifying the Y chromosome, but no signal for DAZ region is present.
1
To whom correspondence should be addressed at: Department of Obstetrics & Gynecology, Division of Reproductive Endocrinology, College of Physicians & Surgeons, Columbia University, 622 West 168th Street, PH 16–28, New York, NY 10032, USA
We thank the families for their cooperation in the study; Dr David C.Page for providing the DAZ cDNA probe and his invaluable help with this manuscript; Dr Kun Ma for providing the MK5 (RBM1) cDNA probe; Dr Peter Vogt for the DNA of Yq microdeleted individuals used to validate our STS PCR methodology; and C.C.Yu and Patricia Lanzano for their invaluable technical assistance.
This study was funded in part by the Columbia Presbyterian Medical Center Office of Clinical Trials House Staff Awards.
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HP40GS vs HP50G
12-11-2015, 01:42 PM
Post: #1
Han Senior Member Posts: 1,881 Joined: Dec 2013
HP40GS vs HP50G
Are these machines similar enough to where flashing one with the other's OS would be feasible?
Graph 3D | QPI | SolveSys
12-11-2015, 03:46 PM
Post: #2
Marcus von Cube Senior Member Posts: 760 Joined: Dec 2013
RE: HP40GS vs HP50G
(12-11-2015 01:42 PM)Han Wrote: Are these machines similar enough to where flashing one with the other's OS would be feasible?
Almost. The 40GS has a lesser display (smaller resolution). It might work because the 48GS has the same display and runs the 50G software. You might be able to find one of these.
Marcus von Cube
Wehrheim, Germany
http://www.mvcsys.de
http://wp34s.sf.net
http://mvcsys.de/doc/basic-compare.html
12-11-2015, 04:58 PM
Post: #3
Massimo Gnerucci Senior Member Posts: 2,378 Joined: Dec 2013
RE: HP40GS vs HP50G
(12-11-2015 03:46 PM)Marcus von Cube Wrote: the 48GS has the same display and runs the 50G software.
Never seen a 48gs.
Greetings,
Massimo
-+×÷ ↔ left is right and right is wrong
12-11-2015, 05:18 PM (This post was last modified: 12-11-2015 05:22 PM by Claudio L..)
Post: #4
Claudio L. Senior Member Posts: 1,830 Joined: Dec 2013
RE: HP40GS vs HP50G
(12-11-2015 03:46 PM)Marcus von Cube Wrote:
(12-11-2015 01:42 PM)Han Wrote: Are these machines similar enough to where flashing one with the other's OS would be feasible?
Almost. The 40GS has a lesser display (smaller resolution). It might work because the 48GS has the same display and runs the 50G software. You might be able to find one of these.
A well known online retailer has them, but $10 more expensive than the 50g... It's the same processor, emulating the same saturn environment, so it's very likely that roms can be exchanged. The 50g ROM can detect the smaller screen of the 48GII, so it might be able to detect this one too. However, you have also only 256 kb of ram instead of 512 kb. Assuming it runs the same emulator, you will have only 128 kb for user ram. The opposite might be more interesting: try the 40gs firmware on a 50g and see if it works. In any case, half the RAM, no SD card and no infrared for$10 more is not exactly the best deal.
EDIT: HPGCC 1.1 used to run on the 39g and 40g, which were also the same hardware, but non-flashable roms.
12-11-2015, 06:42 PM
Post: #5
Marcus von Cube Senior Member Posts: 760 Joined: Dec 2013
RE: HP40GS vs HP50G
(12-11-2015 04:58 PM)Massimo Gnerucci Wrote:
(12-11-2015 03:46 PM)Marcus von Cube Wrote: the 48GS has the same display and runs the 50G software.
Never seen a 48gs.
I think it's named 48gII, not 48gs. I have to find mine and have a look.
Marcus von Cube
Wehrheim, Germany
http://www.mvcsys.de
http://wp34s.sf.net
http://mvcsys.de/doc/basic-compare.html
12-11-2015, 08:09 PM
Post: #6
Massimo Gnerucci Senior Member Posts: 2,378 Joined: Dec 2013
RE: HP40GS vs HP50G
(12-11-2015 06:42 PM)Marcus von Cube Wrote:
(12-11-2015 04:58 PM)Massimo Gnerucci Wrote: Never seen a 48gs.
I think it's named 48gII, not 48gs. I have to find mine and have a look.
Ah, alright then. I thought of a never released model (to go hunting for)... :)
Greetings,
Massimo
-+×÷ ↔ left is right and right is wrong
12-11-2015, 09:08 PM
Post: #7
Han Senior Member Posts: 1,881 Joined: Dec 2013
RE: HP40GS vs HP50G
Thanks everyone for the replies. I had hoped to try out newRPL on one of these, but there's no point if it is more expensive for less hardware.
Graph 3D | QPI | SolveSys
12-11-2015, 09:49 PM
Post: #8
Claudio L. Senior Member Posts: 1,830 Joined: Dec 2013
RE: HP40GS vs HP50G
(12-11-2015 09:08 PM)Han Wrote: Thanks everyone for the replies. I had hoped to try out newRPL on one of these, but there's no point if it is more expensive for less hardware.
Well, you should've started with that!
NewRPL has the RAM size hard-coded, as well as the size of the screen. Changes required would be minimal and in theory it will work perfectly fine on a 40gs, after the proper changes are implemented.
The stock ROM, however, can auto-detect the different models and adapts to the different screen sizes and ram.
12-14-2015, 06:12 AM
Post: #9
cyrille de brébisson Senior Member Posts: 1,047 Joined: Dec 2013
RE: HP40GS vs HP50G
Hello,
The 4 calcs, 39, 40, 48 and 50 (ARM based) have virtually the same HW.
The differences lies in the screen size and (but I am not 100% sure) the Ram size.
What I do not remember is if there is a detection mechanism to let the SW detect which of the 4 HW it is running on.
If this is the case, flashing the code from one onto the other will not work.
Cyrille
Although I work for the HP calculator group, the views and opinions I post here are my own. I do not speak for HP.
12-15-2015, 03:06 AM
Post: #10
Claudio L. Senior Member Posts: 1,830 Joined: Dec 2013
RE: HP40GS vs HP50G
(12-14-2015 06:12 AM)cyrille de brébisson Wrote: Hello,
The 4 calcs, 39, 40, 48 and 50 (ARM based) have virtually the same HW.
The differences lies in the screen size and (but I am not 100% sure) the Ram size.
What I do not remember is if there is a detection mechanism to let the SW detect which of the 4 HW it is running on.
If this is the case, flashing the code from one onto the other will not work.
Cyrille
I vaguely remember Saturn assembler pseudo-opcodes like ISBIGAPPLE? or something like that, used to detect different hardware versions/screen sizes. It could at least detect between 49G+ and regular 49G (large vs small screen).
I don't know if the 39 series had something similar.
12-15-2015, 05:45 AM
Post: #11
cyrille de brébisson Senior Member Posts: 1,047 Joined: Dec 2013
RE: HP40GS vs HP50G
Hello,
The question is: ISBIGAPPLE is based on a HW difference or a compile time setting.
I am quite sure that, in the case of the 39/40, there is a HW detection mechanism (to make sure that the CAS can not be SW enabled by changing a flag)...
But that is honnestly all I remember.
Cyrille
Although I work for the HP calculator group, the views and opinions I post here are my own. I do not speak for HP.
12-15-2015, 06:06 AM
Post: #12
sat1410 Junior Member Posts: 11 Joined: Aug 2015
RE: HP40GS vs HP50G
At least one person tried to convert a 39gs to a 48gII, documented in Chinese here. It appears to be a non-trivial effort, especially considering the price for use 49g+/50g units is not significantly higher than the lesser units.
I think there was at least one more topic on a non-English-language forum that described the process in more detail (I recall that there was a photo of the donor 39gs reassembled and identifying as a 48gII onscreen). The author of that article mentioned that the 39gs was quite inexpensive since many were being sold as surplus without without packaging or manuals in China at the time.
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### C - Candy
##### Languages: C, C++, Java, Tiger, ... (details)
Little Charlie is a nice boy addicted to candies. He is even a subscriber to All Candies Magazine and was selected to participate in the International Candy Picking Contest.
In this contest a random number of boxes containing candies are disposed in $M$ rows with $N$ columns each (so, there are a total of $M \times N$ boxes). Each box has a number indicating how many candies it contains.
The contestant can pick a box (any one) and get all the candies it contains. But there is a catch (there is always a catch): when choosing a box, all the boxes from the rows immediately above and immediately below are emptied, as well as the box to the left and the box to the right of the chosen box. The contestant continues to pick a box until there are no candies left.
The figure bellow illustrates this, step by step. Each cell represents one box and the number of candies it contains. At each step, the chosen box is circled and the shaded cells represent the boxes that will be emptied. After eight steps the game is over and Charlie picked $10 + 9 + 8 + 3 + 7 + 6 + 10 + 1 = 54$ candies.
For small values of $M$ and $N$, Charlie can easily find the maximum number of candies he can pick, but when the numbers are really large he gets completely lost. Can you help Charlie maximize the number of candies he can pick?
#### Input
The input contains several test cases. The first line of a test case contains two positive integers $M$ and $N$ $(1 \leq M \times N \leq 10^5)$, separated by a single space, indicating the number of rows and columns respectively. Each of the following $M$ lines contains $N$ integers separated by single spaces, each representing the initial number of candies in the corresponding box. Each box will have initially at least $1$ and at most $10^3$ candies.
The end of input is indicated by a line containing two zeroes separated by a single space.
The input must be read from standard input.
#### Output
For each test case in the input, your program must print a single line, containing a single value, the integer indicating the maximum number of candies that Charlie can pick.
The output must be written to standard output.
#### Sample test(s)
Input
5 5 1 8 2 1 9 1 7 3 5 2 1 2 10 3 10 8 4 7 9 1 7 1 3 1 6 4 4 10 1 1 10 1 1 1 1 1 1 1 1 10 1 1 10 2 4 9 10 2 7 5 1 1 5 0 0
Output
54 40 17
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# 8 Parameters
The variety of shapes of the nine pattern-book functions means that, often, one or another will be suitable for the modeling situation in hand.
Combining the functions to create a greater diversity of shapes is the subject of Chapter Section 9.
Even if the shape of the function used is appropriate, the pattern still needs to be “adjusted” so that the units of output and input are well matched to the phenomenon being modeled. Let’s consider data from the outbreak of COVID-19 as an example. Figure 8.1 shows, day-by-day, the number of officially confirmed COVID-19 cases as the in the US in March 2020.
During the outbreak, case numbers increased with time. As time went on, the rate of case-number increase itself grew faster and faster. This is the same pattern provided by the exponential function.
Alongside the case-number data Figure 8.1 shows the function $$\text{cases}(t) \equiv e^t$$ plotted as a $$\color{magenta}{\text{magenta}}$$ curve.
There is an obvious mismatch between the data and the function $$e^t$$. Does this mean the COVID pattern is not exponential?
This chapter will introduce how modelers stretch and shift the individual patter-book functions so that they can be used in models of real-world situations such as the outbreak of COVID-19.
## 8.1 Matching numbers to quantities
The coordinate axes in Figure 8.1 represent quantities. On the horizontal axis is time, measured in days. The vertical axis is denominated in “10000 cases,” meaning that the numbers on the vertical scale should be multiplied by 10000 to get the number of cases.
The exponential function takes as input a pure number and produces an output that is also a pure number. This is true for all the pattern-book functions. Since the graph axes don’t show pure numbers, it is no surprise then that the pattern-book exponential function doesn’t align with the COVID case data.
Recall that pure numbers, like 17.32, do not have units. Quantities, on the other hand, usually do have units, as in 17.3 days or 34 meters.
If we want the input to the model function $$\text{cases}(t)$$ to be denominated in days, we will have to convert $$t$$ to a pure pure number (e.g. 10, not “10 days”) before the quantity is handed off as the argument to $$\exp()$$. We do this by introducing a parameter.
In every case, these parameters are arranged to translate a with-units quantity into a pure number suitable as an input to the pattern-book function. Similarly, parameters will translate the pure-number output from the pattern-book function into a quantity with units.
The standard parameterization for the exponential function is $$e^{kt}$$. The parameter $$k$$ will be a quantity with units of “per-day.” Suppose we set $$k=0.2$$ per day. Then $$k\, t{\LARGE\left.\right|}_{t=10 days} = 2$$. This “2” is a pure number because the units on the 0.2 (“per day”) and on the 10 (days) cancel out: $0.2\, \text{day}^{-1} \cdot 10\, \text{days} = 2\ .$ The use of a parameter like $$k$$ does more than handle the formality of converting input quantities into pure numbers. Having a choice for $$k$$ allows us to stretch or compress the function to align with the data. Figure 8.2 plots the modeling version of the exponential function to the COVID-case data:
## 8.2 Parallel scales
At the heart of how we use the pattern-book functions to model the relationship between quantities is the idea of conversion between one scale and another. Consider these everyday objects: a thermometer and a ruler.
Each object presents a read-out of what’s being measured—temperature or length—on two different scales. At the same time, the objects provide a way to convert one scale to another.
A function gives the output for any given input. We represent the input value as a position on a number line—which we call an “axis”—and the output as a position on another output line, almost always drawn perpendicular to one another. But the two number lines can just as well be parallel to one another. To evaluate the function, find the input value on the input scale and read off the corresponding output.
We can translate the correspondance between one scale and the other into the form of a straight-line function. For instance, if we know the temperature in Fahrenheit ($$^\circ$$F) and want to convert it to Celsius ($$^\circ C$$) we have the following function: $C(F) \equiv {\small\frac{5}{9}}(F-32)\ .$ Similarly, converting inches to centimeters can be accomplished with $\text{cm(inches)} \equiv 2.54 \, (\text{inches}-0)\ .$ Both of these scale conversion functions have the form of the straight-line function, which can be written as $f(x) \equiv a x + b\ \ \ \text{or, equivalently as}\ \ \ \ f(x) \equiv a(x-x_0)\ ,$ where $$a$$, $$b$$, and $$x_0$$ are parameters.
In Section Section 8.3, we will use the $$ax + b$$ form of scale conversion, to scale the input to pattern-book functions, but we could equally well have used $$a(x-x_0)$$.
In Section Section 8.4 we will introduce a second scale conversion function, for the output from pattern-book functions. That scaling will also be in the form of a straight-line function: $$A x + B$$. The use of the lower-case parameter names ($$a$$, $$b$$) versus the upper-case parameter names ($$A$$, $$B$$) will help us distinguish the two different uses for scale conversion, namely input scaling versus output scaling.
## 8.3 Input scaling
Figure 8.3 is based on the data frame RI-tide which is a minute-by-minute record of the tide level in Providence, Rhode Island (USA) for the period April 1 to 5, 2010. The level variable is measured in meters; the hour variable gives the time of the measurement in hours after midnight at the start of April 1.
The pattern-book $$\sin()$$ and the function $$\color{magenta}{\text{level}}\color{blue}{(hour)}$$ have similar shapes, so it seems reasonable to model the tide data as a sinusoid. However, the scale of the axes is different on the two graphs.
To model the tide with a sinusoid, we need to modify the sinusoid to change the scale of the input and output. First, let’s look at how to accomplish the input scaling. Specifically, we want the pure-number input $$t$$ to the sinusoid be a function of the quantity $$hour$$. Our framework for this re-scaling is the straight-line function. We will replace the pattern-book input $$t$$ with a function $t(\color{blue}{hour}) \equiv a\, \color{blue}{hour} + b\ .$
The challenge is to find values for the parameters $$a$$ and $$b$$ that will transform the $$\color{blue}{\mathbf{\text{blue}}}$$ horizontal axis into the black horizontal axis, like this:
By comparing the two axes, we can estimate that $$\color{blue}{10} \rightarrow 4$$ and $$\color{blue}{100} \rightarrow 49$$. With these two coordinate points, we can find the straight-line function that turns $$\color{blue}{\mathbf{\text{blue}}}$$ into black by plotting the coordinate pairs $$(\color{blue}{0},1)$$ and $$(\color{blue}{100}, 51)$$ and finding the straight-line function that connects the points.
You can calculate for yourself that the function that relates $$\color{blue}{\mathbf{\text{blue}}}$$ to black is $t(\color{blue}{time}) = \underbrace{\frac{1}{2}}_a \color{blue}{time} \underbrace{-1\LARGE\strut}_b$
Replacing the pure number $$t$$ as the input to pattern-book $$\sin(t)$$ with the transformed $$\frac{1}{2} \color{blue}{time}$$ we get a new function: $g(\color{blue}{time}) \equiv \sin\left(\strut {\small\frac{1}{2}}\color{blue}{time} - 1\right)\ .$ Figure 11.6 plots $$g()$$ along with the actual tide data.
## 8.4 Output scaling
Just as the natural input needs to be scaled before it reaches the pattern-book function, so the output from the pattern-book function needs to be scaled before it presents a result suited for interpreting in the real world.
The overall result of input and output scaling is to tailor the pattern-book function so that it is ready to be used in the real world.
Let’s return to Figure 11.6 which shows that the function $$g(\color{blue}{time})$$, which scales the input to the pattern-book sinusoid, has a much better alignment to the tide data. Still, the vertical axes of the two graphs in the figure are not the same.
This is the job for output scaling, which takes the output of $$g(\color{blue}{time})$$ (bottom graph) and scales it to match the $$\color{magenta}{level}$$ axis on the top graph. That is, we seek to align the black vertical scale with the $$\color{magenta}{\mathbf{\text{magenta}}}$$ vertical scale. To do this, we note that the range of the $$g(\color{blue}{time})$$ is -1 to 1, whereas the range of the tide-level is about 0.5 to 1.5. The output scaling will take the straight-line form ${\color{magenta}{\text{level}}}({\color{blue}{time}}) = A\, g({\color{blue}{time}}) + B$ or, in graphical terms
We can figure out parameters $$A$$ and $$B$$ by finding the straight-line function that connects the coordinate pairs $$(-1, \color{magenta}{0.5})$$ and $$(1, \color{magenta}{1.5})$$ as in Figure 8.7.
You can confirm for yourself that the function that does the job is ${\color{magenta}{\text{level}}} = 0.5 g({\color{blue}{time}}) + 1\ .$
Putting everything together, that is, scaling both the input to pattern-book $$\sin()$$ and the output from pattern-book $$\sin()$$, we get
${\color{magenta}{\text{level}}}({\color{blue}{time}}) = \underbrace{0.5}_A \sin\left(\underbrace{\small\frac{1}{2}}_a {\color{blue}{time}} \underbrace{-1}_b\right) + \underbrace{1}_B$
## 8.5 A procedure for building models
We’ve been using pattern-book functions as the intermediaries between input scaling and output scaling, using this format.
$f(x) \equiv A e^{ax + b} + B\ .$ We can use the other pattern-book functions—the gaussian, the sigmoid, the logarithm, the power-law functions—in the same way. That is, the basic framework for modeling is this:
$\text{model}(x) \equiv A\, {g_{pattern\_book}}(ax + b) + B\ ,$ where $$g_{pattern\_book}()$$ is one of the pattern-book functions. To construct a basic model, you task has two parts:
1. Pick the specific pattern-book function whose shape resembles that of the relationship you are trying to model. For instance, we picked $$e^x$$ for modeling COVID cases versus time (at the start of the pandemic). We picked $$\sin(x)$$ for modeling tide levels versus time.
2. Find numerical values for the parameters $$A$$, $$B$$, $$a$$, and $$b$$. In Chapter ?chap-fitting-by-eye we will show you some ways to make this part of the task easier.
It is remarkable that models of a very wide range of real-world relationships between pairs of quantities can be constructed by picking one of a handful of functions, then scaling the input and the output. As we move on to other Blocks in MOSAIC Calculus, you will see how to generalize this to potentially complicated relationships among more than two quantities. That is a big part of the reason you’re studying calculus.
## 8.6 Other formats for scaling
Often, modelers choose to use input scaling in the form $$a (x - x_0)$$ rather than $$a x + b$$. The two are completely equivalent when $$x_0 = - b/a$$. The choice between the two forms is largely a matter of convention. But almost always the output scaling is written in the format $$A y + B$$.
For the COVID case-number data shown in Figure 8.2, we found that a reasonable match to the data can be had by input- and output-scaling the exponential: $\text{cases}(t) \equiv \underbrace{573}_A e^{\underbrace{0.19}_a\ t}\ .$
You might wonder why the parameters $$B$$ and $$b$$ aren’t included in the model. One reason is that cases and the exponential function already have the same range: zero and upwards. So there is no need to shift the output with a parameter B.
Another reason has to do with the algebraic properties of the exponential function. Specifically, $e^{a x + b}= e^b e^{ax} = {\cal A} e^{ax}$ where $${\cal A} \equiv e^b$$.
In the case of exponentials, writing the input scaling in the form $$e^{a(x-x_0)}$$ can provide additional insight.
A bit of symbolic manipulation of the model can provide some additional insight. As you know, the properties of exponentials and logarithms are such that $A e^{at} = e^{\log(A)} e^{at} = e^{a t + \log(A)} = e^{a\left(\strut t + \log(A)/a\right)} = e^{a(t-t_0)}\ ,$ where $t_0 = - \log(A)/a = - \log(593)/0.19 = -33.6\ .$ You can interpret $$t_0$$ as the starting point of the pandemic. When $$t = t_0$$, the model output is $$e^{k 0} = 1$$: the first case. According to the parameters we matched to the data for March, the pandemic’s first case would have happened about 33 days before March 1, which is late January. We know from other sources of information, the outbreak began in late January. It is remarkable that even though the curve was constructed without any data from January or even February, the data from March, translated through the curve-fitting process, pointed to the start of the outbreak. This is a good indication that the exponential form for the model is fundamentally correct.
## 8.7 Parameterization conventions
There are conventions for the symbols used for input-scaling parameterization of the pattern-book functions. Knowing these conventions makes it easier to read and assimilate mathematical formulas. In several cases, there is more than one conventional option. For instance, the sinusoid has a variety of parameterization forms that get used depending on which feature of the function is easiest to measure. ?tbl-param that are used in practice.
Some standard forms of input scaling parameterizations
Function Written form Parameter 1 Parameter 2
Exponential $$e^{kt}$$ $$k$$ Not used
Exponential $$e^{t/\tau}$$ $$\tau$$ “time constant” Not used
Exponential $$2^{t/\tau_2}$$ $$\tau_2$$ “doubling time” Not used
Exponential $$2^{-\tau_{1/2}}$$ $$-\tau_{1/2}$$ “half life” Not used
Power-law $$[x - x_0]^p$$ $$x_0$$ x-intercept exponent
Sinusoid $$\sin\left(\frac{2 \pi}{P} (t-t_0)\right)$$ $$P$$ “period” $$t_0$$ “time shift”
Sinusoid $$\sin(\omega t + \phi)$$ $$\omega$$ “angular frequency” $$\phi$$ “phase shift”
Sinusoid $$\sin(2 \pi \omega t + \phi)$$ $$\omega$$ “frequency” $$\phi$$ “phase shift”
Gaussian dnorm(x, mean, sd) “mean” (center) sd “standard deviation”
Sigmoid pnorm(x, mean, sd) “mean” (center) sd “standard deviation”
Straight-line $$mx + b$$ $$m$$ “slope” $$b$$ “y-intercept”
Straight-line $$m (x-x_0)$$ $$m$$ “slope” $$x_0$$ “center”
## 8.8 Drill
Part 1 What is the period of the function $$\sin(6\pi t)$$?
1/3 1/2 2 3 6
Part 2 What is the period of $$g(t)$$? $g(t) \equiv \frac{5}{\sin(2 \pi t)}$
1. 1
2. 5
3. $$2 \pi/5$$
4. $$5/2\pi$$
5. $$g(t)$$ isn’t periodic.
Part 3 What is the period of $$g(t)$$? $g(t) \equiv \text{dnorm}\left(\frac{2\pi}{5}(t-3)\right)$
1. 1
2. 5
3. $$2 \pi/5$$
4. $$5/2\pi$$
5. $$g(t)$$ isn’t periodic.
Part 4 One of the following choices is the standard deviation of the function graphed in Figure 8.8. Which one?
0 1 2 3 4
Part 5 What is the value of the parameter “mean” for the function shown in Figure 11.11?
1. -2
2. -1
3. 0.5
4. 1
5. 2
6. “mean” is not a parameter of this function.
Part 6 What is the value of the parameter “sd” for the function shown in ?fig-m06-01?
1. -2
2. -1
3. 0.5
4. 1
5. 2
6. “sd” is not a parameter of this function.
Part 7 What is the value of the parameter “mean” for the function shown in ?fig-m06-02?
1. -2
2. -1
3. 0.5
4. 1
5. 2
6. “mean” is not a parameter of this function.
Part 8 What is the value of the parameter “sd” for the function shown in ?fig-m06-02
1. -2
2. -1
3. 0.5
4. 1
5. 2
6. “sd” is not a parameter of this function.
## 8.9 Exercises
#### Exercise 8.01
Each of the following plots shows a basic modeling function whose input scaling has the form $$x - x_0$$. Your job is to figure out from the graph what is the numerical value of $$x_0$$. (Hint: For simplicity, $$x_0$$ in the questions will always be an integer.)
Part A In plot (A), what is $$x_0$$?
-2 -1 0 1 2
Part B In plot (B), what is $$x_0$$?
-2 -1 0 1 2
Part C In plot (C), what is $$x_0$$?
-2 -1 0 1 2
Part D In plot (D), what is $$x_0$$?
-2 -1 0 1 2
Part E In plot (E), what is $$x_0$$?
-2 -1 0 1 2
#### Exercise 8.02
Each of the graphs shows two horizontal scales, one drawn on the edge graphics frame (black) and one drawn slighly above that (blue). Which horizontal scale (black or blue) corresponds to the pattern-book function shown in the graph?
Part A For graph (A), which scale corresponds to the pattern-book function?
black blue neither both
Part B For graph (B), which scale corresponds to the pattern-book function?
black blue neither both
Part C For graph (C), which scale corresponds to the pattern-book function?
black blue neither both
#### Exercise 8.03
Find the straight-line function that will give the value on the bottom (black) scale for each point $$x$$ on the top (blue) scale. The function will take the top(blue)-scale reading as input and produce the bottom(black)-scale reading as output, that is: $\text{black}(x) \equiv a (x - x_0)$
Part A For Graph A, which function maps blue $$x$$ to the value on the black scale?
$$\frac{1}{3} x$$$$3\, x$$$$x + 3$$$$x - 3$$
Part B For Graph B, which function maps blue $$x$$ to the value on the black scale?
$$-\frac{2}{3}\,x$$$$\frac{3}{2} x$$$$\frac{2}{3} x$$$$-\frac{3}{2}x$$
Part C For Graph C, which function maps blue $$x$$ to the value on the black scale?
$$\frac{1}{2}(x - 2)$$$$3\, x$$$$2\,x$$$$2\,(x + 2)$$
Part D For Graph D, which function maps blue $$x$$ to the value on the black scale?
1. $$\frac{2}{3} (x + 3)$$
2. $$\frac{3}{2} (x - 3)$$
3. $$\frac{3}{2} (x+1)$$
4. $$\frac{3}{2}(x - 2)$$
#### Exercise 8.04
The graph shows a linear combination of two sinusoids, one of period 0.6 and the other of period 2. There is also a baseline shift. That is, the graph shows the function:
$A_1 \sin\left(\frac{2\pi}{2}t\right) + A_2 \sin\left(\frac{2\pi}{0.6} (t-.3)\right) + A_3$
Part A What is $$A_3$$?
-4 -2 0 2 4
Part B What is $$A_1$$?
0 1 2 3.5
Part C What is $$A_2$$?
0 1 2 3.5
#### Exercise 8.05
The Bargain Basement store wants to sell its goods quickly. Consequently, they reduce each product’s price $$P$$ by 5% per day.
Part A If a jacket costs $80 today, how much will it cost in $$t$$ days? $$P = 80 - 5t$$$$P = 80 - 4t$$$$P = 80 - 0.05t$$$$P = 80 (0.05)^t$$$$P = 80 (0.95)^t$$ You will need to use an R console to answer the next question. A hint: the answer is related to the answer from the previous question. Remember, to raise a number to a power, you can use an expression like 0.95^7. Part B You decided that you like the$80 jacket, but you have a budget of only \$60. How long should you wait before coming back to the Bargain Basement store.?
3 days 4 days 5 days 6 days
Part C The answer to the first question is an exponential function, even if at first it doesn’t look like it. Which of these is the same function but written in the standard $$e^{kt}$$ format?
1. $$80 \exp( \ln(0.95) t)$$
2. $$0.95 \exp(80 t)$$
3. $$80 \exp(-\ln(0.95) t)$$
4. $$0.95 \exp(\ln(80) t)$$
#### Exercise 8.06
The three functions created by the statements below are different in important ways. Explain what those differences are.
f1 <- makeFun(sin((2*pi/P)*t) ~ t)
f2 <- makeFun(sin((2*pi/6)*t) ~ t)
f3 <- makeFun(sin((2*pi/P)*t) ~ t, P=6)
#### Exercise 8.07
Watch this movie showing the growth of a colony of E. coli. Each rod is one bacterium.
Bacteria exhibit exponential growth under optimal conditions. In general, if the rate of growth depends on some quantity (here bacteria) then the exponential is the best first guess at a model. In the movie, notice that the rate of expansion depends on the number of bacteria present; the more E.coli, the faster the rate of growth. This is true for any exponential process: the instantaneous rate of growth or decay depends on the amount currently present.
If the experiment were continued indefinitely, the number of bacteria would eventually outgrow the petri dish or deplete their food source. When this happens, we say the bacteria have approached the carrying capacity of their environment. When the population is constrained in this way, a sigmoid would be a more appropriate model to start your modeling process. So, the deciding factor between exponential and sigmoid really depends upon whether 1) we assume a constrained or unconstrained environment, and 2) we let the bacteria reach the carrying capacity of the petri dish or not.
1. Is there any obvious sign that the bacteria are reaching the carrying capacity of their environment before the end of the movie?
2. Estimate the doubling time of the number of bacteria as they are growing exponentially. Do this by figuring out how long it takes the area of the colony to double (roughly). Hint: You’ll need to use the time marker in the bottom left corner of the movie.
3. Estimate the doubling time in another way, by observing an individual bacterium. At any point in the movie, choose a bacterium at random. Watch it until it splits in two then, immediately, note the time. Watch some more until one of the two children split. The time difference between the mother’s split and the child’s split is the doubling time.
4. Compare the doubling time for a mother in the center of a large colony to the doubling time of a mother on the edge of the colony. Is there any clear sign that growth in the more crowded part of the colony is slower than in the suburbs?
Credit: Math 141Z/142Z 2021-2022 development team.
#### Exercise 8.08
Here are four frames from a movie showing (through a microscope) the growth of E. coli. bacteria.
1. In each frame, count the number of bacteria.
2. Construct a data frame recording the time stamp and the number of bacteria in each frame. The unit of observation is a frame. You can use a command like this, replacing the count with your own numbers:
Ecoli <- tibble::tribble(
~ time, ~ count,
100, 36,
150, 289,
200, 1683,
250, 2945
)
1. Make a point plot of the number of bacteria versus time. Use linear, logarithmic or semi-logarithmic axes as most appropriate to show a simple pattern.
A. Which type of axes shows the pattern most simply?
B. Is the pattern most consistent with linear growth, exponential growth, or power-law growth.
C. From your graph, find the parameter that describes the growth rate: - If linear growth, the slope of the line (give units) - If exponential growth, the doubling time (give units) - If power law, the exponent (which will not have units).
#### Exercise 8.09
The diagram shows how the intensity of light from the sun depends on distance $$r$$. Wikipedia link The intensity is the number of photons per unit area. Imagining each red line to be the path followed by one photon, the intensity can be calculated by the area of the surfaces at distance $$r$$, $$2r$$, and $$3r$$.
Part A Which of these functional forms best models intensity $$\cal I$$ as a function of distance $$r$$?
1. Proportional: $$\cal I(r)\equiv ar+b$$
2. Power-law: $$\cal I(r)\equiv Ar^p$$
3. Exponential $$\cal I(r)\equiv Ae^{kr}+C$$
4. Sine: $$\cal I(r)\equiv A\sin \left(\frac{2\pi}{p}(r-r_0)\right)+B$$
5. Sigmoid $$\cal I(r)\equiv A\cdot pnorm(r,mean,sd)+B$$
6. Gaussian $$\cal I(r)\equiv A\cdot dnorm(r,mean,sd)+B$$
#### Exercise 8.10
The performance $$p$$ of a worker depends on the level of stimulation/stress $$s$$ imposed by the task. This phenomenon has come to be known as the Yerkes-Dodson Stress Performance Curve, and you’ve probably experienced this yourself. If a task is not stimulating enough people become inactive/bored and performance is negatively impacted. If tasks are over stimulating (stressful), people become anxious, fatigued, and burn-out. The overall pattern is shown by the diagram.
Part A Which of these functional forms best imitates the Yerkes-Dodson stress performance curve?
1. Proportional: $$p(s) \equiv as+b$$
2. Power-law: $$p(s) \equiv As^p$$
3. Exponential $$p(s) \equiv Ae^{kt}+C$$
4. Sine: $$p(s) \equiv A\sin\left(\frac{2\pi}{p}(t-t_0)\right)+B$$
5. Sigmoid $$p(s) \equiv A\cdot pnorm(s,mean,sd)+B$$
6. Gaussian $$p(s) \equiv A\cdot dnorm(s,mean,sd)+B$$
A manager must balance workloads between too much and too little stimulation to get peak performance out of each team member.
#### Exercise 8.11
The graph shows the proportion $$P$$ of the US cell-phone users who own a smart phone as a function of the year $$y$$.
As a rule, when a quantity grows exponentially but is ultimately limited to some maximum level, the sigmoid is the choice for modeling. The proportion of smartphone owners grew exponentially during the early 2000’s. As the number of smartphones increased, the broader familiarity with and advertisement of smartphones also increased, which help sustain this exponential growth. However, adoption has slowed as smartphone penetration reaches the maximum carrying capacity. In other words, once everyone has a smartphone, the proportion of smartphone owners cannot increase—everyone already owns a smartphone. So, eventually the exponential growth must taper-off. According to several datasets, this inflection point occurred sometime between 2013 and 2014. This behavior is visible in the graphic below showing US smartphone penetration between Jan 2005 and Oct 2020 with raw data shown from 2010 to 2015.
Part A During the initial, exponential phase of smartphone penetration, what was the doubling time for penetration? (Note that the horizontal axis labels have 1/4 year inbetween them.)
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Case weights are positive numeric values that may influence how much each data point has during the preprocessing. There are a variety of situations where case weights can be used.
## Details
tidymodels packages differentiate how different types of case weights should be used during the entire data analysis process, including preprocessing data, model fitting, performance calculations, etc.
The tidymodels packages require users to convert their numeric vectors to a vector class that reflects how these should be used. For example, there are some situations where the weights should not affect operations such as centering and scaling or other preprocessing operations.
The types of weights allowed in tidymodels are:
• Frequency weights via hardhat::frequency_weights()
• Importance weights via hardhat::importance_weights()
More types can be added by request.
For recipes, we distinguish between supervised and unsupervised steps. Supervised steps use the outcome in the calculations, this type of steps will use frequency and importance weights. Unsupervised steps don't use the outcome and will only use frequency weights.
There are 3 main principles about how case weights are used within recipes. First, the data set that is passed to the recipe() function should already have a case weights column in it. This column can be created beforehand using hardhat::frequency_weights() or hardhat::importance_weights(). Second, There can only be 1 case weights column in a recipe at any given time. Third, You can not modify the case weights column with most of the steps or using the update_role() and add_role() functions.
These principles ensure that you experience minimal surprises when using case weights, as the steps automatically apply case weighted operations when supported. The printing method will additionally show which steps where weighted and which steps ignored the weights because they were of an incompatible type.
frequency_weights(), importance_weights()
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# Two definitions of modules in monoidal category
The standard definition of a (left) simplicial module $V$ over some simplicial algebra $A$ is the map of simplicial vector spaces $A\otimes V\to V$ that gives the usual modules component-wise. Here $\otimes$ denotes the component-wise tensor product of simplicial vector spaces (it gives to the category $sVect$ of simplicial vector spaces a symmetric monoidal structure).
But there is a natural structure of simplicial algebra on $sHom(V,V)$ (the internal $Hom$ induced by the product $\otimes$). So we can also define structure of a simplicial module over $A$ on $V$ by fixing a morphism of simplicial algebras $A\to sHom(V,V)$.
I think we can do it in more generality, for any symmetric monoidal category $C$ with internal $Hom$. If we have a monoid $A$ in $C$ and some object $V\in C$, then we can define the structure of $A$-module on $V$ in two different ways: via $A\otimes V\to V$ and $A\to \underline{Hom}(V,V)$, where $\underline{Hom}$ is the internal $Hom$.
The question is: are these two definitions equivalent for any symmetric monoidal category with internal $\underline{Hom}$? If not, is it true for simplicial vector spaces?
Thank you very much!
• If your $\otimes$ and $\underline{\mathrm{Hom}}$ are related by the standard adjunction, then there is a canonical bijection between morphisms $A \otimes V \to V$ and morphisms $A \to \underline{\mathrm{Hom}}(V, V)$. So it just amounts to checking that the axioms for one version translate to the axioms for the other version. Oct 31 '13 at 21:41
• Yes, I know that. But this is far from being obvious (to me) that the axioms actually match up. Maybe I am just being stupid, and there is easy to see. I tried to write it down, but I didn't get anywhere. So I thought maybe there is some more high-powered way to say that everything works. Oct 31 '13 at 21:47
• I'd say this is better for math.stackexchange Oct 31 '13 at 21:47
• Sasha, you may find useful first to check it for abelian groups, where objects are sets and morphisms are maps, and then to rephrase everything in a categorical language Oct 31 '13 at 21:56
I will write $[B, C]$ instead of $\underline{\mathrm{Hom}}(B, C)$. Recall the tensor–hom adjunction: $$\mathrm{Hom}(A \otimes B, C) \cong \mathrm{Hom}(A, [B, C])$$ Thus there is a canonical bijection between morphisms $\alpha : A \otimes V \to V$ and $\tilde{\alpha} : A \to [V, V]$. Let us show that $\alpha$ is an $A$-action on $V$ if and only if $\tilde{\alpha}$ is a monoid homomorphism. For simplicity I will work in a strict monoidal category.
• The unit axiom for $\alpha$ says, $\alpha \circ (e \otimes \mathrm{id}_V) = \mathrm{id}_V$; and the the unit axiom for $\tilde{\alpha}$ says, $\tilde{\alpha} \circ e = \eta_V$, where $\eta_V : I \to [V, V]$ is the right adjoint transpose of $\mathrm{id}_V : V \to V$. The naturality of the tensor–hom adjunction in the first variable implies that these two conditions are equivalent.
• The compatibility axiom for $\alpha$ says, $\alpha \circ (m \otimes \mathrm{id}_V) = \alpha \circ (\mathrm{id}_A \otimes \alpha)$; and the compatibility axiom for $\tilde{\alpha}$ says, $\tilde{\alpha} \circ m = \mu_V \circ (\tilde{\alpha} \otimes \tilde{\alpha})$, where $\mu_V : [V, V] \otimes [V, V] \to [V, V]$ is the right adjoint transpose of the composite $$[V, V] \otimes [V, V] \otimes V \xrightarrow{\mathrm{id}_{[V, V]} \otimes \epsilon_{V,V}} [V, V] \otimes V \xrightarrow{\epsilon_{V,V}} V$$ where $\epsilon_{B,C} : [B, C] \otimes B \to C$ is the left adjoint transpose of $\mathrm{id} : [B, C] \to [B, C]$. Now, naturality in the first variable implies the right adjoint transpose of $\alpha \circ (m \otimes \mathrm{id}_V)$ is $\tilde{\alpha} \circ m$, and naturality in the first variable implies the left adjoint transpose of $\mu_V \circ (\tilde{\alpha} \otimes \tilde{\alpha})$ is the following composite, $$A \otimes A \otimes V \xrightarrow{\tilde{\alpha} \otimes \tilde{\alpha} \otimes \mathrm{id}_V} [V, V] \otimes [V, V] \otimes V \xrightarrow{\mathrm{id}_{[V, V]} \otimes \epsilon_{V,V}} [V, V] \otimes V \xrightarrow{\epsilon_{V,V}} V$$ but $\alpha = \epsilon_{V,V} \circ (\tilde{\alpha} \otimes \mathrm{id}_V)$, so the above reduces to $\alpha \circ (\mathrm{id}_A \otimes \alpha)$.
• There must be a cleaner diagrammatic language to phrase such proofs in. I know there's a nice diagrammatic language when we can write $[B, C] \cong B^{\ast} \otimes C$ but I don't know about in general. I ran into this issue while trying to find a diagrammatic language for cartesian closed categories awhile ago. Oct 31 '13 at 22:27
• There are various extensions of the string diagram calculus to handle biclosed monoidal categories, or at least that's what I hear. For cartesian closed categories one could use simply typed $\lambda$-calculus though. Oct 31 '13 at 23:06
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# Asymptotic work statistics of periodically driven Ising chains
Russomanno, Angelo ; Sharma, Shraddha ; Dutta, Amit ; Santoro, Giuseppe E (2015) Asymptotic work statistics of periodically driven Ising chains Journal of Statistical Mechanics: Theory and Experiment, 2015 (8). P08030. ISSN 1742-5468
Full text not available from this repository.
Official URL: http://doi.org/10.1088/1742-5468/2015/08/P08030
Related URL: http://dx.doi.org/10.1088/1742-5468/2015/08/P08030
## Abstract
We study the work statistics of a periodically-driven integrable closed quantum system, addressing in particular the role played by the presence of a quantum critical point. Taking the example of a one-dimensional transverse Ising model in the presence of a spatially homogeneous but periodically time-varying transverse field of frequency ${{\omega}_{0}}$ , we arrive at the characteristic cumulant generating function G(u), which is then used to calculate the work distribution function P(W). By applying the Floquet theory we show that, in the infinite time limit, P(W) converges, starting from the initial ground state, towards an asymptotic steady state value whose small-W behaviour depends only on the properties of the small-wave-vector modes and on a few important ingredients: the time-averaged value of the transverse field, h0, the initial transverse field, ${{h}_{\text{i}}}$ , and the equilibrium quantum critical point ${{h}_{\text{c}}}$ , which we find to generate a sequence of non-equilibrium critical points ${{h}_{*l}}={{h}_{\text{c}}}+l{{\omega}_{0}}/2$ , with l integer. When ${{h}_{\text{i}}}\ne {{h}_{\text{c}}}$ , we find a 'universal' edge singularity in P(W) at a threshold value of ${{W}_{\text{th}}}=2|{{h}_{\text{i}}}-{{h}_{\text{c}}}|$ which is entirely determined by ${{h}_{\text{i}}}$ . The form of that singularity—Dirac delta derivative or square root—depends on h0 being or not at a non-equilibrium critical point h*l. On the contrary, when ${{h}_{\text{i}}}={{h}_{\text{c}}}$ , G(u) decays as a power-law for large u, leading to different types of edge singularity at ${{W}_{\text{th}}}=0$ . Generalizing our calculations to the case in which we initialize the system in a finite temperature density matrix, the irreversible entropy generated by the periodic driving is also shown to reach a steady state value in the infinite time limit.
Item Type: Article Copyright of this article belongs to IOP Publishing. 117180 16 Apr 2021 04:52 16 Apr 2021 04:52
Repository Staff Only: item control page
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What is Relational Algebra and Relational Calculus in DBMS. Вђў for example, the operation вђў since the result of any relational algebra operation is a the (theta-)join r on, 22/03/2011в в· relational algebra, and relational an overview of relational algebra operators and their sql join b on b.empno=a.empno. theta join operator..
## What are the main operations of Relational Algebra
The Relational Model & Relational Algebra math-cs.gordon.edu. Example instances sid sname rating age relational algebra operation! (operator vsometimes called a theta-join., i believe that with the relational operators given we to add sql equivalents to the relational algebra examples since any particular theta-join on r and.
Example instances sid sname rating age relational algebra operation! (operator vsometimes called a theta-join. what is relational algebra and relational of the join operation. the bnf form for the theta-join based 7 relational algebra example 8:
Join processing in relational databases the jom is the only relational algebra operation that allows the this general join is called a thetayвђќoin. the theta relational algebra the relational model consists of the elements: relations, which are made up notice in the generic (theta) join operation,
## street city Inspiring Innovation
The Relational Model & Relational Algebra math-cs.gordon.edu. Relational algebra in relational dbms. apart from these common operations relational algebra is also used for join operations like, outer join; theta join etc. Cse 544: relational operators, sorting wednesday, theta join, semi-join) algebra (on bags) example select city, count(*).
• Relational Algebra TCNJ
• Basic Operations Algebra of Bags Simon Fraser University
• What are the main operations of Relational Algebra
• Relational Algebra TCNJ
• Example instances sid sname rating age relational algebra operation! (operator vsometimes called a theta-join. basic operators relational algebra there are multiple ways to express the same meaning in relational algebra. example: the theta-join r3 <- r1 joinc r2
Example: boolean algebra.! this means that theta-join is a redundant operator! other relational algebra operations. additional relational operations examples of queries in relational algebra theta join each
Relational algebra the relational model consists of the elements: relations, which are made up notice in the generic (theta) join operation, examples of queries in relational algebra relational algebra operations from set theory binary relational operations: join and division (2/2) theta join
Lecture 16: relational algebra monday, may 10, вђ“natural, equi-join, theta join, semi-join, joins r us вђў the join operation in all its variants (eq- вђў relational algebra (theta) join of r and s and denoted: r theta join вђ“ example a join operation with a general join condition (not an
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• Create Account
Posted 09 May 2012 - 01:09 PM
Using
typedef glm::dvec3 Vector3
was my first idea BUT I want things like the cross product to be used as member functions. GLM implements them as
normal functions taking the two vectors as parameters.
But when the inheritance I have suggested does not have any performance penalties I will go that way.
typedef glm::dvec3 Vector3
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You can reinstall internet explorer if there is some problems, you will need windows xp installation CD to complete this process. the procedure as follows.
• Insert windows xp installation CD to CD Rom
• Now go to start and click run
• Type this command rundll32.exe setupapi,InstallHinfSection DefaultInstall 132 c:\windows\inf\ie.inf and press enter.
• Now proceed the installation process
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# Implicit second order method
(Difference between revisions)
# Implicit Second Order Method
The implicit second order method involves the derivatives of the next time level. Due to this reason they are iterative in nature. The second order time integration scheme is given by:
## Algorithm
for r:= 1 step 1 until M do
$\phi ^{n + 1} = {4 \over 3}\phi ^n - {1 \over 3}\phi ^{n - 1} + {2 \over 3}\dot \phi \left( {t_{n + 1} ,\phi ^{n + 1} } \right) \bullet \Delta t$
end (r-loop)
Where M is maximum number of internal iterations.
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# Last name in bibliography, online latex
I found a bunch of threads with similar problems, but I didn't manage to solve mine, because slightly different. I would like to have the bibliography shown as:
Doniger, W.. Splitting the Difference. Chicago: University of Chicago Press, 1999.
W. Doniger, Splitting the Difference. Chicago: University of Chicago Press, 1999.
my code is:
\documentclass[12pt,a4paper,openany]{report}
\usepackage{natbib}
\bibliographystyle{abbrvnat}
\setcitestyle{authoryear,open={(},close={)}}
...
\printbibliography
\bibliography{bib}
\end{document}
Basically I only need to invert the first name and the last name without changing anything else, because it is already in alphabetical order and with the citation style that I need. I use an online version of latex, so I don't know if I can change the packages (and I don't know how to do it). Any idea?
• Most of the code you have shown is standard BibTeX/natbib, but you also call \printbibliography, which is only defined by biblatex. This should get an error message (if you are using Overleaf, see tex.meta.stackexchange.com/q/7898/35864). You probably should drop the call to \printbibliography with this setup. Jun 25 '19 at 18:31
• That said, the name format is defined by your bibliography style (abbrvnat), if you want to change that, you need to modify the bibliography style or change the style. Jun 25 '19 at 18:36
• Thanks @moewe, it works even without the \printbibliography, but I did not understand how to modify the names order. Could you please give an example or a reference link? Jun 25 '19 at 21:32
• I've taken the liberty of deleting the biblatex and citing tags.
– Mico
Jun 25 '19 at 22:16
I suggest you proceed as follows.
• Find the file abbrvnat.bst in your TeX distribution. If you work with an online LaTeX compiler, I suggest you obtain the file from https://www.ctan.org/tex-archive/macros/latex/contrib/natbib/.
• Make a copy of this file and call the copy, say abbrvnat-reverse.bst. Don't edit an original file of the TeX distribution directly.
• Open the file abbrvnat-reverse.bst in a text editor. The editor you use to edit your tex files will do fine.
• In the file abbrvnat-reverse.bst, locate the function format.names. In my copy of this file, the format.names function starts on line 216.
• In this function, locate the following line:
{ s nameptr "{f.~}{vv~}{ll}{, jj}" format.name$'t := Change this line to { s nameptr "{vv~}{ll}{, jj}{, f.}" format.name$ 't :=
• Save the file abbrvnat-reverse.bst either in the directory where your main tex file is located (this instruction applies if you use an online LaTeX compiler) or in a directory that's searched by BibTeX. If you choose the latter method, please also update the filename database of your TeX distribution suitably.
• In your main TeX file, change
\bibliographystyle{abbrvnat}
to
\bibliographystyle{abbrvnat-reverse}
• Finally, perform a full recompile cycle -- LaTeX, BibTeX, and LaTeX twice more -- to fully propagate all changes.
Happy BibTeXing!
• Thanks a lot @Mico , that was very clear and it worked perfectly! :) Jun 26 '19 at 7:08
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# How do you simplify -12^-1?
Mar 16, 2018
-1/12
#### Explanation:
The negative exponent means that the number will go to the denominator.
$- {12}^{-} 1 = - \frac{1}{12}$
The negative still stays.
Mar 16, 2018
$- \frac{1}{12}$
#### Explanation:
$- {12}^{-} 1 = \frac{1}{-} 12 = - \frac{1}{12}$
any number (except 0) raised to the power of a negative exponent becomes the reciprocal (or multiplicative inverse) of that number (e.g. ${5}^{-} 2 = \frac{1}{5} ^ 2 = \frac{1}{25}$).
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\marginpar inside figure environment fails
I am working with marginpar to put some comments in the margin of my document. In one case I would like to put a comment on a caption of a figure, a MWE of which looks like this:
\begin{document}
\begin{figure}
\caption{captiontext \marginpar{marginText}}
\end{figure}
\end{document}
Unfortunately, this fails saying: Argument of \@caption has an extra }
Why is this happening? And how can I fix it/work around it?
• Is there a particular reason why it should be inside the environment? Since marginpar has no mark, it could just go outside of the figure environment! – Andy Nov 12 '14 at 9:12
• @andy marginpar is actually positioned on on the same horizontal level as the position from which you call it, so that is why I would like to use it within the environment – Michiel Nov 12 '14 at 10:23
• It only is positioned at the same level, if there are not many other marginals pushing it, e.g., down. But David Carlisle's answer also addresses this. – Andy Nov 12 '14 at 11:34
Standard marginpars are added by the page breaker so don't work in any kind of box, so a caption or float box on its own would be a enough, a caption inside a float doubly so. Also the caption is written to the list of figures where you probably don't want the note. You could use \caption[..]{...} to have a note-less caption for the lof.
There are implementations of margin notes that do not use the output routine to add the note, for example
\documentclass{article}
\usepackage{marginnote}
\begin{document}
\begin{figure}
XXXXXX
\caption[zzz]{zzzz\marginnote{this}}
\end{figure}
text.. text.. text.. text.. text.. text.. text..
text.. text.. text.. text.. text.. text.. text..
text.. text.. text.. text.. text.. text.. text..
text.. text.. text.. text.. text.. text.. text..
text.. text.. text.. text.. text.. text.. text..
\end{document}
• Great! Works like a charm. Just for future reference: if you do not include the [zzz] part for the caption, this solution actually doesn't work either. So it is not only nice to include it for the list of figures: it is necessary – Michiel Nov 12 '14 at 10:44
• @Michiel oh well you could use \protect\marginnote if you really wanted the note also in the lof rather than use the [] argument – David Carlisle Nov 12 '14 at 10:46
• I see, but I agree with you that you typically don't want that. – Michiel Nov 12 '14 at 10:48
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VS.
# Break vs. Split
Published:
Breakverb
To separate into two or more pieces, to fracture or crack, by a process that cannot easily be reversed for reassembly.
‘If the vase falls to the floor, it might break.’; ‘In order to tend to the accident victim, he will break the window of the car.’;
Divided.
‘Republicans appear split on the centerpiece of Mr. Obama's economic recovery plan.’;
Breakverb
To crack or fracture (bone) under a physical strain.
‘His ribs broke under the weight of the rocks piled on his chest.’; ‘She broke her neck.’; ‘He slipped on the ice and broke his leg.’;
Having the middle group equal to the direct product of the others.
Breakverb
(transitive) To divide (something, often money) into smaller units.
‘Can you break a hundred-dollar bill for me?’; ‘The wholesaler broke the container loads into palettes and boxes for local retailers.’;
(of coffee) Comprising half decaffeinated and half caffeinated espresso.
Breakverb
(transitive) To cause (a person or animal) to lose spirit or will; to crush the spirits of.
‘Her child's death broke Angela.’; ‘Interrogators have used many forms of torture to break prisoners of war.’; ‘The interrogator hoped to break her to get her testimony against her accomplices.’; ‘You have to break an elephant before you can use it as an animal of burden.’;
Divided so as to be done or executed part at one time or price and part at another time or price.
Breakverb
(intransitive) To be crushed, or overwhelmed with sorrow or grief.
‘My heart is breaking.’;
Given in sixteenths rather than the usual eighths.
$10\frac\left\{3\right\}\left\{16\right\}$ is a split quotation.’;
Breakverb
(transitive) To interrupt; to destroy the continuity of; to dissolve or terminate.
‘I've got to break this habit I have of biting my nails.’; ‘to break silence; to break one's sleep; to break one's journey’; ‘I had won four games in a row, but now you've broken my streak of luck.’;
(London stock exchange) Designating ordinary stock that has been divided into preferred ordinary and deferred ordinary.
Breakverb
(transitive) To ruin financially.
‘The recession broke some small businesses.’;
Splitnoun
A crack or longitudinal fissure.
Breakverb
(transitive) To violate, to not adhere to.
‘When you go to Vancouver, promise me you won't break the law.’; ‘He broke his vows by cheating on his wife.’; ‘break one's word’; ‘Time travel would break the laws of physics.’;
Splitnoun
A breach or separation, as in a political party; a division.
Breakverb
To pass the most dangerous part of the illness; to go down, in terms of temperature.
‘Susan's fever broke at about 3 AM, and the doctor said the worst was over.’;
Splitnoun
A piece that is split off, or made thin, by splitting; a splinter; a fragment.
Breakverb
To end.
‘The forecast says the hot weather will break by midweek.’;
Splitnoun
(leather manufacture) One of the sections of a skin made by dividing it into two or more thicknesses.
Breakverb
To begin; to end.
‘We ran to find shelter before the storm broke.’; ‘Around midday the storm broke, and the afternoon was calm and sunny.’;
Splitnoun
A maneuver of spreading or sliding the feet apart until the legs are flat on the floor 180 degrees apart, either sideways to the body or with one leg in front and one behind, thus lowering the body completely to the floor in an upright position.
Breakverb
To arrive.
‘Morning has broken.’; ‘The day broke crisp and clear.’;
Splitnoun
A split-finger fastball.
‘He’s got a nasty split.’;
Breakverb
To render (a game) unchallenging by altering its rules or exploiting loopholes or weaknesses in them in a way that gives a player an unfair advantage.
‘Changing the rules to let white have three extra queens would break chess.’; ‘I broke the RPG by training every member of my party to cast fireballs as well as use swords.’;
Splitnoun
(bowling) A result of a first throw that leaves two or more pins standing with one or more pins between them knocked down.
Breakverb
To stop, or to cause to stop, functioning properly or altogether.
‘On the hottest day of the year the refrigerator broke.’; ‘Did you two break the trolley by racing with it?’;
Splitnoun
A split shot or split stroke.
Breakverb
To cause (some feature of a program or piece of software) to stop functioning properly; to cause a regression.
‘Adding 64-bit support broke backward compatibility with earlier versions.’;
Splitnoun
A dessert or confection resembling a banana split.
Breakverb
(transitive) To cause (a barrier) to no longer bar.
‘break a seal’;
Splitnoun
A unit of measure used for champagne or other spirits: 18.75 centiliter or one quarter of a standard .75 liter bottle. Commercially comparable to 1/20th (US) gallon, which is 1/2 of a fifth.
Breakverb
(specifically) To cause the shell of (an egg) to crack, so that the inside (yolk) is accessible.
Splitnoun
A bottle of wine containing 0.375 liters, half the volume of a standard .75 liter bottle; a demi.
Breakverb
(specifically) To open (a safe) without using the correct key, combination, or the like.
Splitnoun
(athletics) The elapsed time at specific intermediate points in a race.
‘In the 3000m race, his 800m split was 1:45.32’;
Breakverb
(transitive) To destroy the arrangement of; to throw into disorder; to pierce.
‘The cavalry were not able to break the British squares.’;
Splitnoun
(video games) The elapsed time at specific intermediate points in a speedrun.
Breakverb
To collapse into surf, after arriving in shallow water.
Splitnoun
(construction) A tear resulting from tensile stresses.
Breakverb
(intransitive) To burst forth; to make its way; to come into view.
Splitnoun
(gambling) A division of a stake happening when two cards of the kind on which the stake is laid are dealt in the same turn.
Breakverb
(intransitive) To interrupt or cease one's work or occupation temporarily.
‘Let's break for lunch.’;
Splitnoun
(music) A recording containing songs by multiple artists.
Breakverb
(transitive) To interrupt (a fall) by inserting something so that the falling object does not (immediately) hit something else beneath.
‘He survived the jump out the window because the bushes below broke his fall.’;
Splitverb
Of something solid, to divide fully or partly along a more or less straight line.
‘He has split his lip.’;
Breakverb
To disclose or make known an item of news, etc.
‘The newsman wanted to break a big story, something that would make him famous.’; ‘I don't know how to break this to you, but your cat is not coming back.’; ‘In the latest breaking news...’; ‘When news of their divorce broke, ...’;
Splitverb
(intransitive) Of something solid particularly wood, to break along the grain fully or partly along a more or less straight line.
Breakverb
To become audible suddenly.
Splitverb
(transitive) To share; to divide.
‘We split the money among three people.’;
Breakverb
(transitive) To change a steady state abruptly.
‘His coughing broke the silence.’; ‘His turning on the lights broke the enchantment.’; ‘With the mood broken, what we had been doing seemed pretty silly.’;
Splitverb
(slang) To leave.
‘Let's split this scene and see if we can find a real party.’;
Breakverb
To suddenly become.
‘Things began breaking bad for him when his parents died.’; ‘The arrest was standard, when suddenly the suspect broke ugly.’;
Splitverb
To separate or break up.
‘Did you hear Dick and Jane split? They'll probably get a divorce.’;
Breakverb
(intransitive) Of a male voice, to become deeper at puberty.
Splitverb
To factor into linear factors.
Breakverb
(intransitive) Of a voice, to alter in type due to emotion or strain: in men generally to go up, in women sometimes to go down; to crack.
‘His voice breaks when he gets emotional.’;
Splitverb
To be broken; to be dashed to pieces.
Breakverb
(transitive) To surpass or do better than (a specific number), to do better than (a record), setting a new record.
‘He broke the men's 100-meter record.’; ‘I can't believe she broke 3 under par!’; ‘The policeman broke sixty on a residential street in his hurry to catch the thief.’;
Splitverb
To burst out laughing.
Breakverb
:
Splitverb
To divulge a secret; to betray confidence; to peach.
Breakverb
To win a game (against one's opponent) as receiver.
‘He needs to break serve to win the match.’;
Splitverb
(sports) In athletics (esp. baseball), when both teams involved in a doubleheader each win one game and lose another game.
‘Boston split with Philadelphia in a doubleheader, winning the first game 3-1 before losing 2-0 in the nightcap.’;
Breakverb
To make the first shot; to scatter the balls from the initial neat arrangement.
‘Is it your or my turn to break?’;
Splitverb
To divide lengthwise; to separate from end to end, esp. by force; to divide in the direction of the grain or layers; to rive; to cleave; as, to split a piece of timber or a board; to split a gem; to split a sheepskin.
‘Cold winter split the rocks in twain.’;
Breakverb
To remove one of the two men on (a point).
Splitverb
To burst; to rupture; to rend; to tear asunder.
‘A huge vessel of exceeding hard marble split asunder by congealed water.’;
Breakverb
To demote, to reduce the military rank of.
Splitverb
To divide or break up into parts or divisions, as by discord; to separate into parts or parties, as a political party; to disunite.
Breakverb
(transitive) To end (a connection), to disconnect.
‘The referee ordered the boxers to break the clinch.’; ‘The referee broke the boxers' clinch.’; ‘I couldn't hear a thing he was saying, so I broke the connection and called him back.’;
Splitverb
To divide or separate into components; - often used with up; as, to split up sugar into alcohol and carbonic acid.
Breakverb
To demulsify.
Splitverb
To part asunder; to be rent; to burst; as, vessels split by the freezing of water in them.
Breakverb
To counter-attack
Splitverb
To be broken; to be dashed to pieces.
‘The ship splits on the rock.’;
Breakverb
To lay open, as a purpose; to disclose, divulge, or communicate.
Splitverb
To separate into parties or factions.
Breakverb
(intransitive) To become weakened in constitution or faculties; to lose health or strength.
Splitverb
To burst with laughter.
‘Each had a gravity would make you split.’;
Breakverb
To fail in business; to become bankrupt.
Splitverb
To divulge a secret; to betray confidence; to peach.
Breakverb
(transitive) To destroy the strength, firmness, or consistency of.
‘to break flax’;
Splitverb
To divide one hand of blackjack into two hands; - a strategy allowed to a player when the first two cards dealt to the player have the same value.
Breakverb
(transitive) To destroy the official character and standing of; to cashier; to dismiss.
Splitverb
To leave; to depart (from a place or gathering); as, let's split.
Breakverb
(intransitive) To make an abrupt or sudden change; to change the gait.
‘to break into a run or gallop’;
Splitnoun
A crack, rent, or longitudinal fissure.
Breakverb
To fall out; to terminate friendship.
Splitnoun
A breach or separation, as in a political party; a division.
Breaknoun
An instance of breaking something into two or more pieces.
‘The femur has a clean break and so should heal easily.’;
Splitnoun
A piece that is split off, or made thin, by splitting; a splinter; a fragment.
Breaknoun
A physical space that opens up in something or between two things.
‘The sun came out in a break in the clouds.’; ‘He waited minutes for a break in the traffic to cross the highway.’;
Splitnoun
One of the sections of a skin made by dividing it into two or more thicknesses.
Breaknoun
A rest or pause, usually from work.
‘Let’s take a five-minute break.’;
Splitnoun
A division of a stake happening when two cards of the kind on which the stake is laid are dealt in the same turn.
Breaknoun
A short holiday.
‘a weekend break on the Isle of Wight’;
Splitnoun
Any of the three or four strips into which osiers are commonly cleft for certain kinds of work; - usually in pl.
Breaknoun
A temporary split with a romantic partner.
‘I think we need a break.’;
Splitnoun
Short for Split shot or split stroke.
Breaknoun
An interval or intermission between two parts of a performance, for example a theatre show, broadcast, or sports game.
Splitnoun
The feat of going down to the floor so that the legs extend in a straight line, either with one on each side or with one in front and the other behind.
Breaknoun
A significant change in circumstance, attitude, perception, or focus of attention.
‘big break’; ‘lucky break, bad break’;
Splitnoun
A small bottle (containing about half a pint) of some drink; - so called as containing half the quantity of the customary smaller commercial size of bottle; also, a drink of half the usual quantity; a half glass.
Breaknoun
The beginning (of the morning).
‘at the break of day’;
Splitnoun
The substitution of more than one share of a corporation's stock for one share. The market price of the stock usually drops in proportion to the increase in outstanding shares of stock. The split may be in any ratio, as, a two-for-one split; a three-for-two split.
Breaknoun
An act of escaping.
‘make a break for it, for the door’; ‘It was a clean break.’; ‘prison break’;
Splitnoun
The division by a player of one hand of blackjack into two hands, allowed when the first two cards dealt to a player have the same value; the player who chooses to split is obliged to increase the amount wagered by placing a sum equal to the original bet on the new hand thus created. See split{6}, v.i.
Breaknoun
The separation between lines or paragraphs of a written text.
Divided; cleft.
Breaknoun
A change, particularly the end of a spell of persistent good or bad weather.
Divided deeply; cleft.
Breaknoun
:
Divided so as to be done or executed part at one time or price and part at another time or price; - said of an order, sale, etc.
Breaknoun
(tennis) A game won by the receiving player(s).
Splitnoun
extending the legs at right angles to the trunks (one in front and the other in back)
Breaknoun
The first shot in a game of billiards
Splitnoun
a bottle containing half the usual amount
Breaknoun
(snooker) The number of points scored by one player in one visit to the table
Splitnoun
a promised or claimed share of loot or money;
‘he demanded his split before they disbanded’;
Breaknoun
(soccer) The counter-attack
Splitnoun
a lengthwise crack in wood;
‘he inserted the wedge into a split in the log’;
Breaknoun
(surfing) A place where waves break (that is, where waves pitch or spill forward creating white water).
‘The final break in the Greenmount area is Kirra Point.’;
Splitnoun
an opening made forcibly as by pulling apart;
‘there was a rip in his pants’; ‘she had snags in her stockings’;
Breaknoun
(dated) A large four-wheeled carriage, having a straight body and calash top, with the driver's seat in front and the footman's behind.
Splitnoun
an old Croatian city on the Adriatic Sea
Breaknoun
(equitation) A sharp bit or snaffle.
Splitnoun
a dessert of sliced fruit and ice cream covered with whipped cream and cherries and nuts
Breaknoun
(music) A short section of music, often between verses, in which some performers stop while others continue.
‘The fiddle break was amazing; it was a pity the singer came back in on the wrong note.’;
Splitnoun
(tenpin bowling) a divided formation of pins left standing after the first bowl;
‘he was winning until he got a split in the tenth frame’;
Breaknoun
(music) The point in the musical scale at which a woodwind instrument is designed to overblow, that is, to move from its lower to its upper register.
‘Crossing the break smoothly is one of the first lessons the young clarinettist needs to master.’;
Splitnoun
an increase in the number of outstanding shares of a corporation without changing the shareholders' equity;
‘they announced a two-for-one split of the common stock’;
Breaknoun
(music) A section of extended repetition of the percussion break to a song, created by a hip-hop DJ as rhythmic dance music.
Splitnoun
the act of rending or ripping or splitting something;
‘he gave the envelope a vigorous rip’;
Breakverb
To strain apart; to sever by fracture; to divide with violence; as, to break a rope or chain; to break a seal; to break an axle; to break rocks or coal; to break a lock.
Splitnoun
division of a group into opposing factions;
‘another schism like that and they will wind up in bankruptcy’;
Breakverb
To lay open as by breaking; to divide; as, to break a package of goods.
Splitverb
separate into parts or portions;
‘divide the cake into three equal parts’; ‘The British carved up the Ottoman Empire after World War I’;
Breakverb
To lay open, as a purpose; to disclose, divulge, or communicate.
‘Katharine, break thy mind to me.’;
Splitverb
separate or cut with a tool, such as a sharp instrument;
‘cleave the bone’;
Breakverb
To infringe or violate, as an obligation, law, or promise.
‘Out, out, hyena! these are thy wonted arts . . . To break all faith, all vows, deceive, betray.’;
Splitverb
discontinue an association or relation; go different ways;
‘The business partners broke over a tax question’; ‘The couple separated after 25 years of marriage’; ‘My friend and I split up’;
Breakverb
To interrupt; to destroy the continuity of; to dissolve or terminate; as, to break silence; to break one's sleep; to break one's journey.
‘Go, release them, Ariel;My charms I'll break, their senses I'll restore.’;
Splitverb
go one's own away; move apart;
‘The friends separated after the party’;
Breakverb
To destroy the completeness of; to remove a part from; as, to break a set.
Splitverb
break open or apart suddenly;
‘The bubble burst’;
Breakverb
To destroy the arrangement of; to throw into disorder; to pierce; as, the cavalry were not able to break the British squares.
being divided or separated;
‘split between love and hate’;
Breakverb
To shatter to pieces; to reduce to fragments.
‘The victim broke in pieces the musical instruments with which he had solaced the hours of captivity.’;
having been divided; having the unity destroyed;
‘Congress...gave the impression of...a confusing sum of disconnected local forces’; ‘a league of disunited nations’; ‘a fragmented coalition’; ‘a split group’;
Breakverb
To exchange for other money or currency of smaller denomination; as, to break a five dollar bill.
broken or burst apart longitudinally;
‘after the thunderstorm we found a tree with a split trunk’; ‘they tore big juicy chunks from the heart of the split watermelon’;
Breakverb
To destroy the strength, firmness, or consistency of; as, to break flax.
having a long rip or tear;
‘a split lip’;
Breakverb
To weaken or impair, as health, spirit, or mind.
‘An old man, broken with the storms of state.’;
(especially of wood) cut or ripped longitudinally with the grain;
‘we bought split logs for the fireplace’;
Breakverb
To diminish the force of; to lessen the shock of, as a fall or blow.
‘I'll rather leap down first, and break your fall.’;
Breakverb
To impart, as news or information; to broach; - with to, and often with a modified word implying some reserve; as, to break the news gently to the widow; to break a purpose cautiously to a friend.
Breakverb
To tame; to reduce to subjection; to make tractable; to discipline; as, to break a horse to the harness or saddle.
‘Why, then thou canst not break her to the lute?’;
Breakverb
To destroy the financial credit of; to make bankrupt; to ruin.
‘With arts like these rich Matho, when he speaks,Attracts all fees, and little lawyers breaks.’;
Breakverb
To destroy the official character and standing of; to cashier; to dismiss.
‘I see a great officer broken.’;
Breakverb
To come apart or divide into two or more pieces, usually with suddenness and violence; to part; to burst asunder.
Breakverb
To open spontaneously, or by pressure from within, as a bubble, a tumor, a seed vessel, a bag.
‘Else the bottle break, and the wine runneth out.’;
Breakverb
To burst forth; to make its way; to come to view; to appear; to dawn.
‘The day begins to break, and night is fled.’; ‘And from the turf a fountain broke,and gurgled at our feet.’;
Breakverb
To burst forth violently, as a storm.
‘The clouds are still above; and, while I speak,A second deluge o'er our head may break.’;
Breakverb
To open up; to be scattered; to be dissipated; as, the clouds are breaking.
‘At length the darkness begins to break.’;
Breakverb
To become weakened in constitution or faculties; to lose health or strength.
‘See how the dean begins to break;Poor gentleman! he droops apace.’;
Breakverb
To be crushed, or overwhelmed with sorrow or grief; as, my heart is breaking.
Breakverb
To fall in business; to become bankrupt.
‘He that puts all upon adventures doth oftentimes break, and come to poverty.’;
Breakverb
To make an abrupt or sudden change; to change the gait; as, to break into a run or gallop.
Breakverb
To fail in musical quality; as, a singer's voice breaks when it is strained beyond its compass and a tone or note is not completed, but degenerates into an unmusical sound instead. Also, to change in tone, as a boy's voice at puberty.
Breakverb
To fall out; to terminate friendship.
‘To break upon the score of danger or expense is to be mean and narrow-spirited.’; ‘Fear me not, man; I will not break away.’; ‘He had broken down almost at the outset.’; ‘This radiant from the circling crowd he broke.’;
Breaknoun
An opening made by fracture or disruption.
Breaknoun
An interruption of continuity; change of direction; as, a break in a wall; a break in the deck of a ship.
Breaknoun
An interruption; a pause; as, a break in friendship; a break in the conversation.
Breaknoun
An interruption in continuity in writing or printing, as where there is an omission, an unfilled line, etc.
‘All modern trash isSet forth with numerous breaks and dashes.’;
Breaknoun
The first appearing, as of light in the morning; the dawn; as, the break of day; the break of dawn.
Breaknoun
A large four-wheeled carriage, having a straight body and calash top, with the driver's seat in front and the footman's behind.
Breaknoun
A device for checking motion, or for measuring friction. See Brake, n. 9 & 10.
Breaknoun
See Commutator.
Breaknoun
some abrupt occurrence that interrupts;
‘the telephone is an annoying interruption’; ‘there was a break in the action when a player was hurt’;
Breaknoun
an unexpected piece of good luck;
‘he finally got his big break’;
Breaknoun
(geology) a crack in the earth's crust resulting from the displacement of one side with respect to the other;
‘they built it right over a geological fault’;
Breaknoun
a personal or social separation (as between opposing factions);
‘they hoped to avoid a break in relations’;
Breaknoun
a pause from doing something (as work);
‘we took a 10-minute break’; ‘he took time out to recuperate’;
Breaknoun
the act of breaking something;
‘the breakage was unavoidable’;
Breaknoun
a time interval during which there is a temporary cessation of something
Breaknoun
breaking of hard tissue such as bone;
‘it was a nasty fracture’; ‘the break seems to have been caused by a fall’;
Breaknoun
the occurrence of breaking;
‘the break in the dam threatened the valley’;
Breaknoun
the opening shot that scatters the balls in billiards or pool
Breaknoun
(tennis) a score consisting of winning a game when your opponent was serving;
‘he was up two breaks in the second set’;
Breaknoun
an act of delaying or interrupting the continuity;
‘it was presented without commercial breaks’;
Breaknoun
a sudden dash;
‘he made a break for the open door’;
Breaknoun
any frame in which a bowler fails to make a strike or spare;
‘the break in the eighth frame cost him the match’;
Breaknoun
an escape from jail;
‘the breakout was carefully planned’;
Breakverb
terminate;
‘She interrupted her pregnancy’; ‘break a lucky streak’; ‘break the cycle of poverty’;
Breakverb
become separated into pieces or fragments;
‘The figurine broke’; ‘The freshly baked loaf fell apart’;
Breakverb
destroy the integrity of; usually by force; cause to separate into pieces or fragments;
‘He broke the glass plate’; ‘She broke the match’;
Breakverb
render inoperable or ineffective;
‘You broke the alarm clock when you took it apart!’;
Breakverb
ruin completely;
Breakverb
act in disregard of laws and rules;
‘offend all laws of humanity’; ‘violate the basic laws or human civilization’; ‘break a law’;
Breakverb
move away or escape suddenly;
‘The horses broke from the stable’; ‘Three inmates broke jail’; ‘Nobody can break out--this prison is high security’;
Breakverb
scatter or part;
‘The clouds broke after the heavy downpour’;
Breakverb
force out or release suddenly and often violently something pent up;
‘break into tears’; ‘erupt in anger’;
Breakverb
prevent completion;
‘stop the project’; ‘break off the negociations’;
Breakverb
enter someone's property in an unauthorized manner, usually with the intent to steal or commit a violent act;
‘Someone broke in while I was on vacation’; ‘They broke into my car and stole my radio!’;
Breakverb
make submissive, obedient, or useful;
‘The horse was tough to break’; ‘I broke in the new intern’;
Breakverb
fail to agree with; be in violation of; as of rules or patterns;
‘This sentence violates the rules of syntax’;
Breakverb
surpass in excellence;
‘She bettered her own record’; ‘break a record’;
Breakverb
make known to the public information that was previously known only to a few people or that was meant to be kept a secret;
‘The auction house would not disclose the price at which the van Gogh had sold’; ‘The actress won't reveal how old she is’; ‘bring out the truth’; ‘he broke the news to her’;
Breakverb
come into being;
‘light broke over the horizon’; ‘Voices broke in the air’;
Breakverb
stop operating or functioning;
‘The engine finally went’; ‘The car died on the road’; ‘The bus we travelled in broke down on the way to town’; ‘The coffee maker broke’; ‘The engine failed on the way to town’; ‘her eyesight went after the accident’;
Breakverb
interrupt a continued activity;
Breakverb
make a rupture in the ranks of the enemy or one's own by quitting or fleeing;
‘The ranks broke’;
Breakverb
curl over and fall apart in surf or foam, of waves;
‘The surf broke’;
Breakverb
lessen in force or effect;
‘soften a shock’; ‘break a fall’;
Breakverb
be broken in;
‘If the new teacher won't break, we'll add some stress’;
Breakverb
come to an end;
‘The heat wave finally broke yesterday’;
Breakverb
vary or interrupt a uniformity or continuity;
‘The flat plain was broken by tall mesas’;
Breakverb
cause to give up a habit;
‘She finally broke herself of smoking cigarettes’;
Breakverb
give up;
‘break cigarette smoking’;
Breakverb
come forth or begin from a state of latency;
‘The first winter storm broke over New York’;
Breakverb
happen or take place;
‘Things have been breaking pretty well for us in the past few months’;
Breakverb
cause the failure or ruin of;
‘His peccadilloes finally broke his marriage’; ‘This play will either make or break the playwright’;
Breakverb
invalidate by judicial action;
‘The will was broken’;
Breakverb
discontinue an association or relation; go different ways;
‘The business partners broke over a tax question’; ‘The couple separated after 25 years of marriage’; ‘My friend and I split up’;
Breakverb
assign to a lower position; reduce in rank;
‘She was demoted because she always speaks up’; ‘He was broken down to Sargeant’;
Breakverb
reduce to bankruptcy;
‘My daughter's fancy wedding is going to break me!’; ‘The slump in the financial markets smashed him’;
Breakverb
change directions suddenly
Breakverb
emerge from the surface of a body of water;
‘The whales broke’;
Breakverb
break down, literally or metaphorically;
‘The wall collapsed’; ‘The business collapsed’; ‘The dam broke’; ‘The roof collapsed’; ‘The wall gave in’; ‘The roof finally gave under the weight of the ice’;
Breakverb
do a break dance;
‘Kids were break-dancing at the street corner’;
Breakverb
exchange for smaller units of money;
‘I had to break a \$100 bill just to buy the candy’;
Breakverb
destroy the completeness of a set of related items;
‘The book dealer would not break the set’;
Breakverb
make the opening shot that scatters the balls
Breakverb
separate from a clinch, in boxing;
‘The referee broke the boxers’;
Breakverb
go to pieces;
‘The lawn mower finally broke’; ‘The gears wore out’; ‘The old chair finally fell apart completely’;
Breakverb
break a piece from a whole;
‘break a branch from a tree’;
Breakverb
become punctured or penetrated;
‘The skin broke’;
Breakverb
pierce or penetrate;
Breakverb
be released or become known; of news;
‘News of her death broke in the morning’;
Breakverb
cease an action temporarily;
‘We pause for station identification’; ‘let's break for lunch’;
Breakverb
interrupt the flow of current in;
‘break a circuit’;
Breakverb
undergo breaking;
‘The simple vowels broke in many Germanic languages’;
Breakverb
find a flaw in;
‘break an alibi’; ‘break down a proof’;
Breakverb
find the solution or key to;
‘break the code’;
Breakverb
change suddenly from one tone quality or register to another;
‘Her voice broke to a whisper when she started to talk about her children’;
Breakverb
happen;
‘Report the news as it develops’; ‘These political movements recrudesce from time to time’;
Breakverb
become fractured; break or crack on the surface only;
‘The glass cracked when it was heated’;
Breakverb
of the male voice in puberty;
‘his voice is breaking--he should no longer sing in the choir’;
Breakverb
fall sharply;
‘stock prices broke’;
Breakverb
fracture a bone of;
‘I broke my foot while playing hockey’;
Breakverb
diminish or discontinue abruptly;
‘The patient's fever broke last night’;
Breakverb
weaken or destroy in spirit or body;
‘His resistance was broken’; ‘a man broken by the terrible experience of near-death’;
Breakverb
separate into pieces as a result of a blow, shock, or strain
‘the rope broke with a loud snap’; ‘windows in the street were broken by the blast’;
Breakverb
sustain an injury involving the fracture of a bone or bones in a part of the body
‘what if his leg had broken?’; ‘she had broken her leg in two places’;
Breakverb
cause a cut or graze in (the skin)
‘the bite had scarcely broken the skin’;
Breakverb
make or become inoperative
‘he's broken the video’; ‘the machine has broken and they can't fix it until next week’;
Breakverb
(of the amniotic fluid surrounding a fetus) be discharged when the sac is ruptured in the first stages of labour
‘she realized her waters had broken’;
Breakverb
open (a safe) forcibly.
Breakverb
use (a banknote) to pay for something and receive change out of the transaction
‘she had to break a tenner’;
Breakverb
(of two boxers or wrestlers) come out of a clinch, especially at the referee's command
‘I was acting as referee and telling them to break’;
Breakverb
make the first stroke at the beginning of a game of billiards, pool, or snooker.
Breakverb
unfurl (a flag or sail).
Breakverb
succeed in deciphering (a code)
‘ciphers are easily broken by the new wonder machines’;
Breakverb
disprove (an alibi).
Breakverb
interrupt (a continuity, sequence, or course)
‘the new government broke the pattern of growth’; ‘his concentration was broken by a sound’;
Breakverb
put an end to (a silence) by speaking or making contact
‘it was some time before he broke the silence’;
Breakverb
make a pause in (a journey)
‘we will break our journey in Venice’;
Breakverb
stop proceedings in order to have a pause or vacation
‘at mid-morning they broke for coffee’;
Breakverb
lessen the impact of (a fall)
‘she put out an arm to break her fall’;
Breakverb
disconnect or interrupt (an electric circuit)
‘a multimeter able to measure current without having to break the circuit under test’;
Breakverb
stop oneself from engaging in (a habitual practice)
‘try to break the habit of adding salt at the table’;
Breakverb
surpass (a record)
‘the film broke box office records in the US’;
Breakverb
fail to observe (a law, regulation, or agreement)
‘the council says it will prosecute traders who break the law’; ‘a legally binding contract which can only be broken by mutual consent’;
Breakverb
fail to continue with (a self-imposed discipline)
‘diets started without preparation are broken all the time’;
Breakverb
crush the emotional strength, spirit, or resistance of
‘the idea was to better the prisoners, not to break them’;
Breakverb
(of a person's emotional strength or control) give way
‘her self-control finally broke’;
Breakverb
destroy the power of (a movement or organization)
‘strategies used to break the union’;
Breakverb
destroy the effectiveness of (a strike), typically by moving in other people to replace the striking workers
‘a government threat to use the army to break the strike’;
Breakverb
(of the weather) change suddenly, especially after a fine spell
‘the weather broke and thunder rumbled through a leaden sky’;
Breakverb
(of a storm) begin violently
‘when all were aboard, the storm broke’;
Breakverb
(of dawn or a day) begin as the sun rises
‘dawn was just breaking’;
Breakverb
(of clouds) move apart and begin to disperse
‘on the seventh of September the clouds broke for the first time’;
Breakverb
(of waves) curl over and dissolve into foam
‘the Caribbean sea was breaking gently on the shore’;
Breakverb
(of a person's voice) falter and change tone, due to emotion
‘her voice broke as she relived the experience’;
Breakverb
(of a boy's voice) change in tone and register at puberty
‘after his voice broke, he left the choir’;
Breakverb
(of a vowel) develop into a diphthong, under the influence of an adjacent sound.
Breakverb
(of prices on the stock exchange) fall sharply.
Breakverb
(of news or a scandal) suddenly become public
‘since the news broke I've received thousands of wonderful letters’;
Breakverb
make bad news known to (someone)
‘he was trying to break the terrible news gently to his father’;
Breakverb
(chiefly of an attacking player or team, or of a military force) make a rush or dash in a particular direction
‘Mitchell won possession and broke quickly, allowing Hughes to score’;
Breakverb
(of a bowled cricket ball) change direction on bouncing, due to spin.
Breakverb
(of a ball) rebound unpredictably
‘the ball broke to Craig but his shot rebounded from the post’;
Breaknoun
an interruption of continuity or uniformity
‘the magazine has been published without a break since 1950’;
Breaknoun
an act of separating oneself from a pre-existing state of affairs
‘a break with the past’;
Breaknoun
a change in the weather
‘a week or so may pass without a break in the weather’;
Breaknoun
a change of line, paragraph, or page
‘dotted lines on the screen show page breaks’;
Breaknoun
a change of tone in a person's voice due to emotion
‘there was a break in her voice now’;
Breaknoun
an interruption in an electric circuit.
Breaknoun
the winning of a game against an opponent's serve.
Breaknoun
a pause in work or during an activity or event
‘I need a break from mental activity’; ‘a coffee break’; ‘those returning to work after a career break’;
Breaknoun
an interval during the school day
‘the bell went for break’;
Breaknoun
a short holiday
‘a weekend break in the Cotswolds’;
Breaknoun
a short solo or instrumental passage in jazz or popular music.
Breaknoun
dance music featuring breakbeats.
Breaknoun
a gap or opening
‘the track bends left through a break in the hedge’; ‘he stopped to wait for a break in the traffic’;
Breaknoun
an instance of breaking something, or the point where something is broken
‘he was stretchered off with a break to the leg’;
Breaknoun
a rush or dash in a particular direction, especially by an attacking player or team
‘Norwich scored on a rare break with 11 minutes left’;
Breaknoun
an escape, typically from prison.
Breaknoun
a change in the direction of a bowled ball on bouncing.
Breaknoun
an opportunity or chance, especially one leading to professional success
‘he got his break as an entertainer on a TV music hall show’;
Breaknoun
a consecutive series of successful shots, scoring a specified number of points
‘a break of 83 put him in front for the first time’;
Breaknoun
a player's turn to make the opening shot of a game
‘whose break is it?’;
Breaknoun
a bud or shoot sprouting from a stem.
Breaknoun
former term for breaking cart
Breaknoun
another term for brake
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# Chebyshev’s Inequality
If we have a random variable $X$ and any number $a$, what is the probability that $X \geq a$? If we know the mean of $X$, Markov’s inequality can give an upper bound on the probability that $X$. As it turns out, this upper bound is rather loose, and it can be improved if we know the variance of $X$ in addition to its mean. This result is known as Chebyshev’s inequality after the name of the famous mathematician Pafnuty Lvovich Chebyshev. It was first proved by his student Andrey Markov who provided a proof in 1884 in his PhD thesis (see wikipedia).
How likely is it that the absolute value of a random variable $X$ will be greater than some specific value, say, $a$? The Russian mathematician Andrey Andreyevich Markov proved a simple, yet nice, result which enables us to answer the above question.
Markov’s Inequality states that if $X$ is a random variable with mean $\mu$, and $a>0$ is a positive number, then
$Pr[|X| \geq a] \leq \frac{|\mu|}{a}$
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# What Do We Care About?
## Reliability
Definition: Working correctly under faults and errors.
We need reliability in a system because users can be stupid and malicious and make mistakes. We want the system to be fault-tolerant in these conditions.
### Aside: Fault vs. Failure
A fault is defined as a component of a system not behaving as it is supposed to. A failure is defined as system-wide inability to perform a function. Sometimes, faults in fault-tolerant systems are triggered deliberately so that it is continuously tested.
TL;DR: Faults can happen, but we need to respond to it so that no failures happen.
### Aside: Types of Faults
A hardware fault is when a machine dies, and is largely random. This is often solved by redundancy via RAIDs, dual power supplies, etc. You can actually write software to increase hardware fault-tolerance.
A software fault is often logical. This is somewhat harder because writing software to find bugs in software is non-computable mathematically (it’s not even Turing-recognizable). The only thing we can do is design the system so that errors are propagated up to the developer as soon as possible.
TL;DR: Hardware faults are random and easier to solve than software faults, which are often due to bugs in code.
## Scalability
Definition: How well a system can deal with growing complexity or load.
Load is a very ambiguous concept. It’s simply a metric of things that often add complexity to a system. For example, requests per second, hit rate on a cache, amount of data in a database.
When we increase load, we either want to keep the system the same, and see how the performance is affected, or vice versa. For example, throughput is the amount of things done by a system, and response time is the time required by the system to do a thing. Similar to response time, latency is the time spent waiting for a request to be handled. Latency is factored into response time. Looking at throughput and response time are two sides of the same dice.
TL;DR: To analyze load, you want to analyze how fast the system processes one thing, or how many things a system can process in a given time interval.
### Aside: Statistical Analysis on Response Time
Once enough response time metrics are obtained, we retrieve a discrete distribution. Often people report the average response time, but it doesn’t really take into account how many users are actually experiencing the delay. We often care about this because the users experiencing the highest delay are the ones using the system the most. They often make us the most money.
Instead, we should use percentiles. The 99th percentile are the slowest 1% response times, often due to a large latency. Often, percentiles are used in SLA (service level agreements).
TL;DR: Use percentiles, not the average.
### Aside: How to Deal With Increasing Load
In order to deal with increasing load, one can scale up/vertically by moving to a more powerful machine, or scale out/horizontally by adding more machines. Obviously, the power of a machine and its price is often nonlinear, and we want to be cheap. However, not all systems can be scaled out efficiently. There is a tradeoff dependent on the architecture. Elastic scaling is automatically scaling when detecting increasing load. This can be vertical or horizontal but most often it is horizontal.
TL;DR: To scale, throw more machines at the problem or buy a beefier one.
## Maintainability
Definition: How well the system is able to be maintained and improved.
Noone likes legacy systems because reading other people’s code is hard. Most of the cost of running a system is not during development but during maintenance by an operations team. Let’s save some money:
• Operability: You want the operations team to easily maintain the system.
• Simplicity: You want new engineers to not run away when they try to understand the system.
• Evolvability: Make it easy to add changes to the system.
TL;DR: Design a system with maintainance and extensibility in mind.
### Aside: What Operation Teams Do
Operation teams write a lot of tooling around monitoring, failure investigations, security patching, etc. They are one of the most important teams, but are often overlooked because they wipe the subpar developers’ asses. They often do migration, maintenance, config management, deployment, documentation, etc. It’s a lot of hard work.
TL;DR: Operation teams are unsung heroes who make sure the system is well maintained.
### Aside: How to KISS
By KISS, I meant Keep It Simple, Stupid. When a project gets large, uncontrolled complexity grows roughly quadratically (There are $N^2$ edges in a complete simple undirected graph with $N$ nodes). We want to keep the complexity growth as linear as possible when new components are added.
Sometimes it’s the subpar developer’s fault, but sometimes it’s the customer’s fault. Often times, stupid anti-patterns emerge where the users do something they’re not supposed to and future versions of projects need to keep backwards compatibility. This accidental complexity is there to stay, and that sucks.
To remove complexity, we use abstractions. Basically hide a lot of implementation details under a simple interface. This is surprisingly hard.
TL;DR: Don’t make things too complicated, and prevent users from doing stupid shit by hiding things from them.
### Aside: How to Evolve
This is more of a engineering design process kind of issue. The development team needs to be receptive to constant change due to regulations, scalability concerns, etc. Agile is a complicated development pattern that focuses on iterative development and frequent introspections in order to change specifications on the fly.
TL;DR: Agile is one method to allow for frequently changing specifications and evolution of a project.
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1. ## finding primes
How many primes are in the form 4n^8+1, where n is a positive integer?
*show working n explanation, PLS...
2. Hello
n=5p (where p is an integer) is a necessary condition but I do not know if it is a sufficient condition
If n=5p+1
Then n^8=(5p+1)^8=5P+1
Therefore 4n^8+1=20P+5 is divisible by 5 => not prime
If n=5p+2
Then n^8=(5p+2)^8=5P+2^8=5P+256
Therefore 4n^8+1=20P+1025 is divisible by 5 => not prime
If n=5p+3
Then n^8=(5p+3)^8=5P+3^8=5P+6561
Therefore 4n^8+1=20P+26245 is divisible by 5 => not prime
If n=5p+4
Then n^8=(5p+4)^8=5P+4^8=5P+65536
Therefore 4n^8+1=20P+262145 is divisible by 5 => not prime
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# Math Help - Prove that f(x) is an odd function?
1. ## Prove that f(x) is an odd function?
Prove that $f(x) = \left\{ \begin{array}{cc}x^4\tan \frac{\pi x}{2}, &\mbox{if}\ |x|<1\\x|x|, &\mbox{if}\ |x|\geq 1\end{array}\right.$ is an odd function.
2. We want $f(x)=-f(-x) \: \forall x \in \mathbb{R}$.
If $|x|<1, f(-x) = (-x)^4 \tan(-\pi x/2) = x^4 (-\tan (\pi x/2)) = -f(x)$
If $|x|>1, f(-x) = (-x)|(-x)| = \left\{ \begin{array}{cc}-x|-x|, &\mbox{if}\ x<0\\-x|x|, &\mbox{if}\ x>0\end{array}\right.
$
so that in all cases $-f(-x)=f(x)$
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# Math equation with cases and division signs
\begin{equation*}
\theta({P,\bar{Cl}}) =
\begin{cases}
S(P,\widehat{Cl}) + (S(P,\widehat{Cl}) \times ((\bar{R}-\widehat{R})/\bar{R})) & \widehat{Cl} \textless \bar{Cl} \\
S(P,\widehat{Cl}) - (S(P,\widehat{Cl}) \times ((\widehat{R}-\bar{R})/\bar{R})) & \widehat{Cl} \textgreater \bar{Cl}
\end{cases}
\label(eq: state}
\end{equation*}
How can I make it look more beautiful and legible? I would like to push the division sign below so that it is more legible
any ideas are welcome
• Use \frac{}{} instead of a division line. But this depends one what is more beautiful to you. – dustin Oct 7 '14 at 13:26
• I tried \frac what is the difference between frac and frax or is it a typo? – tandem Oct 7 '14 at 13:26
• I did that then. any other suggestions will be appreciated. – tandem Oct 7 '14 at 13:31
• What does beautiful mean to you? This question is very subjective which makes it hard to answer. Why don't you like \frac? If you use dcases, you will have display style if that is beautiful. I think you need to load mathtools instead of amsmath instead though for dcases. – dustin Oct 7 '14 at 13:33
• I will take a look at some examples on the wiki page in that case. – tandem Oct 7 '14 at 13:38
First of all, I would use displaystyle fractions. Next I would replace the non-extensible \bar with \widebar, borrowed from the mathx(it's in the mathabx bundle, without loading the mathabx package since it changes many symbols.As I'm changing symbols, I also replaced \widehat with mathx's version, with I named varwidehat to not interfere with defaults. Finally I changed Cl into a math operator (Cl), as I suspect it is so. For the moment it looks like the product of the two variables C and l. If I'm wrong, all you have to do is replacing back \Cl with Cl.
To prevent having too large vertical spacing with respect to the surrounding text, due to the fractions, I kill the first line fraction height with a \smash[t]{…} command, and the second line fraction depth with \smash[b]{…}.
\documentclass[11pt,a4paper]{article}
\usepackage{mathtools}
\DeclareMathOperator\Cl{Cl}
\DeclareFontFamily{U}{mathx}{\hyphenchar\font45}
\DeclareFontShape{U}{mathx}{m}{n}{
<-6> mathx5 <6-7> mathx6 <7-8> mathx7
<8-9> mathx8 <9-10> mathx9
<10-12> mathx10 <12-> mathx12
}{}
\DeclareSymbolFont{mathx}{U}{mathx}{m}{n}
\DeclareFontSubstitution{U}{mathx}{m}{n}
\DeclareMathAccent{\varwidehat}{0}{mathx}{"70}
\DeclareMathAccent{\widebar}{0}{mathx}{"73}
\begin{document}
\begin{equation*}
\theta({P,\widebar{\Cl}}) =
\begin{cases}
S(P,\varwidehat{\Cl}) + S(P,\varwidehat{\Cl}) \times \smash[t]{ \dfrac{\widebar{R}-\varwidehat{R}}{\widebar{R}}} &\qquad \varwidehat{\Cl} < \widebar{\Cl} \\[1.5ex]
S(P,\varwidehat{\Cl}) - S(P,\varwidehat{\Cl}) \times \smash[b]{\dfrac{\varwidehat{R}-\widebar{R}}{\widebar{R}}} &\qquad \varwidehat{\Cl} > \widebar{\Cl}
\end{cases}
\label{(eq: state}
\end{equation*}
\end{document}
• Ingenious use of \smash[t]{...} in the first line and \smash[b]{...} in the second. You may want to explain in more detail what these macros do, just in case the OP isn't an experienced user of the amsmath package... – Mico Oct 7 '14 at 14:57
• You're right, I'll do it. – Bernard Oct 7 '14 at 14:59
Your code isn't compilable since it contains macros such as \textgreater and \textless that shouldn't occur in math mode. The argument of a \label command has to be enclosed in curly braces, by the way.
You didn't give any hints as to your criteria for beauty and legibility, so I'll have to use my own. :-) There's nothing terribly wrong with the look once the code's been fixed so that it compiles. (I'm talking purely about the aestetics, of course; I can't tell if the math is correct or not...) I did notice that the overall look is quite cramped. You could use a dcases* environment (provided by the mathtools package) instead of the basic cases environment to provide a bit more whitespace. Next, there's a minor issue with the uneven usage of accents: the "hats" are uniformly "wide", but the "bars" are not. I suggest using \overline for the Cl terms. Third, I'd get rid of two redundant pairs of parentheses on each line. Finally, since there's no point in providing a \label to an unnumbered equation, I'd either comment it out or get rid of it entirely.
\documentclass[12pt, a4paper]{book}
\usepackage{mathtools} % for 'dcases*' environment
\begin{document}
Before:
\begin{equation*}
\theta({P,\bar{Cl}}) =
\begin{cases}
S(P,\widehat{Cl}) + (S(P,\widehat{Cl}) \times ((\bar{R}-\widehat{R})/\bar{R})) & \widehat{Cl} < \bar{Cl} \\
S(P,\widehat{Cl}) - (S(P,\widehat{Cl}) \times ((\widehat{R}-\bar{R})/\bar{R})) & \widehat{Cl} > \bar{Cl}
\end{cases}
\label{eq: state}
\end{equation*}
After:
\begin{equation*}
\theta(P,\overline{Cl}\,) =
\begin{dcases*}
S(P,\widehat{Cl}) + S(P,\widehat{Cl}) \times (\bar{R}-\widehat{R})/\bar{R}
& if $\widehat{Cl} < \overline{Cl}$ \\
S(P,\widehat{Cl}) - S(P,\widehat{Cl}) \times (\widehat{R}-\bar{R})/\bar{R}
& if $\widehat{Cl} > \overline{Cl}$
\end{dcases*}
\end{equation*}
\end{document}
Here are two suggestions that might make the equation easier to read.
Code based on @Bernard s answer.
\documentclass[11pt,a4paper]{article}
\usepackage{mathtools}
\DeclareMathOperator\Cl{Cl}
\DeclareFontFamily{U}{mathx}{\hyphenchar\font45}
\DeclareFontShape{U}{mathx}{m}{n}{
<-6> mathx5 <6-7> mathx6 <7-8> mathx7
<8-9> mathx8 <9-10> mathx9
<10-12> mathx10 <12-> mathx12
}{}
\DeclareSymbolFont{mathx}{U}{mathx}{m}{n}
\DeclareFontSubstitution{U}{mathx}{m}{n}
\DeclareMathAccent{\varwidehat}{0}{mathx}{"70}
\DeclareMathAccent{\widebar}{0}{mathx}{"73}
\begin{document}
\begin{equation*}
\theta({P,\widebar{\Cl}}) =
\begin{cases}
S(P,\varwidehat{\Cl}) \times \left( 1 +
\varwidehat{\Cl} < \widebar{\Cl} \\[3.0ex]
S(P,\varwidehat{\Cl}) \times \left( 1 -
\varwidehat{\Cl} > \widebar{\Cl}
\end{cases}
\label{(eq: state}
\end{equation*}
or perhaps, if this difference has some separate semantic value
Let
%
\begin{equation*}
\Delta = \dfrac{\widebar{R}-\varwidehat{R}}{\widebar{R}}.
\end{equation*}
Then
\begin{equation*}
\theta({P,\widebar{\Cl}}) =
\begin{cases}
S(P,\varwidehat{\Cl}) \times ( 1 + \Delta)
• I think the cases environment would look more appealing if the large parentheses had the same size on both lines. Currently, the parentheses seem to be too small on the first line and too large on the second. – Mico Oct 8 '14 at 21:00
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# 8th European PostgraduateFluid Dynamics Conference
## 6th – 9th July 2016Warsaw, Poland
Back to programme
# Turbulent combustion modelling with OpenFOAM: simulation of the non-premixed syngas flame
Michał T. Lewandowski1, Jacek Pozorski2
1Institute of Fluid-Flow Machinery, Gdańsk University of Technology
2Institute of Fluid-Flow Machinery
Turbulent combustion is a phenomenon of multi-scale nature and its modelling is a challenge, mainly due to turbulence and complex chemical kinetics. The key problem is related to the unknown expression for the mean reaction rate. The modelling based on the physical analysis of the phenomena is needed. Those methods are commonly referred to as combustion models or more precisely turbulence-chemistry interaction models.
In this work, we present numerical simulations of the turbulent non-premixed syngas flame in the configuration of axisymmetric jet also known as Sandia CHN flame B. The assessment of two turbulence-chemistry interaction models has been investigated using opensource software OpenFOAM. The Partially Stirred Reactor model and the Eddy Dissipation Concept have been tested together with different chemical kinetic schemes. Turbulence closure has been obtained with selected versions of the two equation $k-\epsilon$ model. Large Eddy Simulation will also be
Keywords: Combustion, Turbulence, Applied fluid dynamics
Figure 1: Temperature distribution along the axis for CHN flame B obtained with PaSR model FFR chemical kinetic scheme and four different versions of $k-\epsilon$ turbulence
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## Cryptology ePrint Archive: Report 2019/287
Security Evaluation for Snow 2.0-like Stream Ciphers Against Correlation Attacks over Extension Fields
A. N. Alekseychuk and S. M. Koniushok and M. V. Poremskyi
Abstract: We propose a general method for security evaluation of SNOW 2.0-like ciphers against correlation attacks that are built similarly to known attacks on SNOW 2.0. Unlike previously known methods, the method we propose is targeted at security proof and allows obtaining lower bounds for efficiency of attacks from the class under consideration directly using parameters of stream cipher components similarly to techniques for security proofs of block ciphers against linear cryptanalysis. The method proposed is based upon automata-theoretic approach to evaluation the imbalance of discrete functions. In particular, we obtain a matrix representation and upper bounds for imbalance of an arbitrary discrete function being realized by a sequence of finite automata. These results generalize a number of previously known statements on matrix (linear) representations for imbalance of functions having specified forms, and may be applied to security proofs for other stream ciphers against correlation attacks. Application of this method to SNOW 2.0 and Strumok ciphers shows that any of the considered correlation attacks on them over the field of the order 256 has an average time complexity not less than $2^{146.20}$ and $2^{249.40}$ respectively, and requires not less than $2^{142.77}$ and, respectively, $2^{249.38}$ keystream symbols.
Category / Keywords: secret-key cryptography / symmetric cryptography, stream cipher, correlation attack, system of noised linear equations, discrete Fourier transform, proof of security, SNOW 2.0, Strumok.
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# zbMATH — the first resource for mathematics
## Journal of Mathematics. Wuhan University
Short Title: J. Math., Wuhan Univ. Parallel Title: Shuxue Zazhi. Wuhan Daxue Publisher: Wuhan University, Wuchang ISSN: 0255-7797 Online: http://www.oriprobe.com/journals/sxzz.html Comments: Indexed cover-to-cover
Documents indexed: 3,529 Publications (since 1981)
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**CS112 PA2: Basic Shading** ***Due date: Wednesday, Nov 6 by 11:59 pm*** Introduction ==================== The objective of this assignment is to get familiar with the basic geometric transformations that are fundamental to creating any scene. Specifically, you would need to write some code (maybe dozens of lines), and it will take some time to understand every part, so **please start early**. **Software and hardware requirement**: WebGL runs within the browser and is independent of the operating and window systems. You may finish the assignment using any operating system you like (e.g. Windows, MacOS or Linux). **Programming language**: The assignment will be implemented in JavaScript. As we will minimize the use of obscure JavaScript language features, it should not be too difficult for anyone with prior experiences in dynamic languages like Python or familiar with the principles of Object Oriented Programming (like C++ or Java) to get a handle on JavaScript syntax by reading through some of the code in code skeleton. For more formal introductions to JavaScript, checkout the [references](#usefulreferences). **Cooperation and third-party code**: This is an individual programming assignment, and you should implement the code by your own. You may not share final solutions, and you must write up your own work, expressing it in your own words/notation. Third party codes are not allowed. Getting started with the code skeleton ======================================== Download pa2.zip -------------------- Download [pa2.zip](pa2.zip) and extract it any folder you like. You should have five files in the folder: |file name|description| |---|---| |pa2.html |HTML file which shows the WebGL canvas.| |pa2.js |functions used to draw the scene.| |gl-matrix-min.js |utility functions for operating matrices.| |model.js |file describing the models in IFS format.| |trackball-rotator.js|utility functions for rotating the scene using the cursor.| Open pa2.html -------------------- Open *pa2.html* in the extracted folder with Chrome. You will see an error message right now as the vertex and fragment shaders are not implemented. The vertex and fragment shaders are responsible for drawing the scene.  Note -------------------- - For this project, **you will be modifying two files *pa2.html* and *pa2.js***. But try to read and understand what other functions are doing as well as it would help you better implement things. There are lots of comments throughout the codebase and the functions have similar structure as programming assignment 1 for better/easy understanding. - Using the [object tab], a user can select different models (torus, cylinder or sphere). Using the [lighting checkmarks], a user can toggle between different illumination conditions. A user can also adjust the [light position] and [exponent] used for computing specular highlights. There are two tasks which need to be completed in this programming assignment. They are as follows. Implement the lighting function ======================================== TASK 1 ---------- You will be implementing 3 different kinds of illuminations: ambient, diffuse, and specular using the Phong shading model. You will complete both vertex and fragment shader in *pa2.html* in order to draw object. ### Task 1-1 Before starting the project, you need to first know how to compute the light at a single point. Below is a schematic of a point on a surface which is being illuminated by a point light source where: - $L$ = The direction of the light - $N$ = Normal of the point on the surface - $R$ = Reflection of $L$ about $N$ - $V$ = Viewing direction  - Light at a point = Ambient Light + Diffuse Light + Specular Light - Ambient Light = $K_a \times$ diffuse_color - Diffuse Light = $K_d \times$ diffuse_color $\times \max(N\cdot L, 0)$ - Specular Light = $K_s \times$ specular_color $\times \max(R\cdot V, 0)^n$ Use $K_a=0.1, K_d=0.8, K_s=0.4$. $n$ is the specular coefficient, which the user can change. Note, although the above light equation is not complete as it does not account for the distance from light source (light intensity decreases as we move away from the light source), to keep things simple we would only implement above equation in this programming assignment. ### Task 1-2 The next thing you need to know is what are Shaders. Shaders (Vertex, Fragment, etc) are small programs which are executed on the GPU. They are mainly responsible for determining the color of each object. However, they can be used for many different tasks. In this assignment we will be using two shaders namely *vertex* and *fragment* shaders to render of our scene. To write in any shader, you must first know the basics of GLSL. GLSL is a short term for OpenGL Shading Language. This is used to write code into any shader. It is similar to C/C++. Check out the references to understand more on how to write code in a shader. Once you are familiar with GLSL, you can start writing code in the vertex and fragment shader in *pa2.html*.  However, before writing, I would recommend checking out the function **initGL()** in *pa2.js* to see what all variables you will be creating in the shaders. Note, you will be using some of these variables in the vertex shader and some in the fragment shader. Hence, it's important to understand: - What vertex and fragment shaders do. - What these variables are and what you can compute from them.  ### Task 1-3 You will need to implemented each of the lighting separately such that when the user checks ambient or diffuse or specular only those are being implemented. If everything works well, you should be getting results similar to the below images.     TASK 2 ---------- Once you have implement all the three lighting with Phong shading, add two more light positions, one at the bottom of the object and the other on the left of the object. ### Task 2-1 Add two more light positions for the user to select. When the user selects this light position, the scene should be rendered appropriately.  ### Task 2-2 What is happening when the user changes the specular exponent value? Why is this happening. Describe this no more than 3 lines below the scene in the *pa2.html* file. (Feel free to play around with different objects, lighting and light positions to get a sense as to what is happening)  Submission ==================== 1. Make sure your name and ID is filled above the canvas in *pa2.html*. 1. You will need to submit the following files on EEE Canvas in **a zip archive**, please DO NOT submit individual files. - pa2.html - pa2.js - model.js - gl-matrix-min.js - trackball-rotator.js 1. We will grade your work using Google Chrome by default. If your code requires a different browser to work properly, please clarify by including a *readme* file in your zip archive. Grading ==================== 1. (60 pts) Your program should be able to correctly light up all the three objects for all the three lighting scenarios (ambient, diffuse, specular) using the Phong shading. The user should be able to manipulate these scenarios using the lighting checkboxes. 2. (30 pts) You should be successfully able to add two new lighting positions (one at the bottom and one on the left) to the existing codebase. When selected, they should light the objects appropriately. 3. (10 pts) In your own words you should also describe how the specular exponent value affects the final appearance. Useful references ==================== - [WebGL](https://webglfundamentals.org) - [JavaScript 1](https://developer.mozilla.org/en-US/docs/Web/JavaScript), [JavaScript 2](https://google.github.io/styleguide/javascriptguide.xml?showone=Comments#Comments) - [glMatrix](http://glmatrix.net) - [WebGL Shaders and GLSL (GL Shading Language)](https://webglfundamentals.org/webgl/lessons/webgl-shaders-and-glsl.html)
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Under the auspices of the Computational Complexity Foundation (CCF)
REPORTS > KEYWORD > EXPONENTIAL TIME HYPOTHESIS:
Reports tagged with Exponential Time Hypothesis:
TR10-078 | 27th April 2010
Holger Dell, Thore Husfeldt, Martin Wahlén
Exponential Time Complexity of the Permanent and the Tutte Polynomial
Revisions: 1
The Exponential Time Hypothesis (ETH) says that deciding the satisfiability of $n$-variable 3-CNF formulas requires time $\exp(\Omega(n))$. We relax this hypothesis by introducing its counting version #ETH, namely that every algorithm that counts the satisfying assignments requires time $\exp(\Omega(n))$. We transfer the sparsification lemma for $d$-CNF formulas to the counting ... more >>>
TR14-012 | 27th January 2014
Scott Aaronson, Russell Impagliazzo, Dana Moshkovitz
AM with Multiple Merlins
Revisions: 1
We introduce and study a new model of interactive proofs: AM(k), or Arthur-Merlin with k non-communicating Merlins. Unlike with the better-known MIP, here the assumption is that each Merlin receives an independent random challenge from Arthur. One motivation for this model (which we explore in detail) comes from the close ... more >>>
TR14-092 | 22nd July 2014
Mark Braverman, Young Kun Ko, Omri Weinstein
Approximating the best Nash Equilibrium in $n^{o(\log n)}$-time breaks the Exponential Time Hypothesis
The celebrated PPAD hardness result for finding an exact Nash equilibrium in a two-player game
initiated a quest for finding \emph{approximate} Nash equilibria efficiently, and is one of the major open questions in algorithmic game theory.
We study the computational complexity of finding an $\eps$-approximate Nash equilibrium with good social ... more >>>
TR16-195 | 19th November 2016
Pasin Manurangsi
Almost-Polynomial Ratio ETH-Hardness of Approximating Densest $k$-Subgraph
Revisions: 1
In the Densest $k$-Subgraph problem, given an undirected graph $G$ and an integer $k$, the goal is to find a subgraph of $G$ on $k$ vertices that contains maximum number of edges. Even though the state-of-the-art algorithm for the problem achieves only $O(n^{1/4 + \varepsilon})$ approximation ratio (Bhaskara et al., ... more >>>
TR18-103 | 30th April 2018
Zhao Song, David Woodruff, Peilin Zhong
Relative Error Tensor Low Rank Approximation
We consider relative error low rank approximation of tensors with respect to the Frobenius norm. Namely, given an order-$q$ tensor $A \in \mathbb{R}^{\prod_{i=1}^q n_i}$, output a rank-$k$ tensor $B$ for which $\|A-B\|_F^2 \leq (1+\epsilon) {\rm OPT}$, where ${\rm OPT} = \inf_{\textrm{rank-}k~A'} \|A-A'\|_F^2$. Despite much success on obtaining relative error low ... more >>>
ISSN 1433-8092 | Imprint
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# What should be the intuition when working with compactness?
I have a question that may be regarded by many as duplicate since there’s a similar one at MathOverflow. The point is that I think I’m not really getting the idea on compactness. I mean, in $$Rn\mathbb{R}^n$$ the compact sets are those that are closed and bounded, however the guy who answered this question and had his answer accepted says that compactness is some analogue of finiteness.
That’s the first problem: In my intuitive view of finiteness, only boundedness would suffice to say that a certain subset of $$Rn\mathbb{R}^n$$ is in some sense “finite”. On the other hand there’s the other definition of compactness (in terms of covers) which is the one I really need to work with and I cannot see how that definition implies this intuition on finiteness.
Also, I feel it’s pretty strange the covers people use when they want to deal with compact sets. To prove a set is compact I know they must show that for every open cover there’s a finite subcover; the problem is that I can’t see intuitively how one could show this for every cover. Also when trying to disprove compactness the books I’ve read start presenting strange covers that I would have never thought about. I think my real problem is that I didn’t yet get the intuition on compactness.
So, what intuition should we have about compact sets in general and how should we really put this definition to use?
Can someone provide some reference that shows how to understand the process of proving (and disproving) compactness?
The following story may or may not be helpful. Suppose you live in a world where there are two types of animals: Foos, which are red and short, and Bars, which are blue and tall. Naturally, in your language, the word for Foo over time has come to refer to things which are red and short, and the word for Bar over time has come to refer to things which are blue and tall. (Your language doesn’t have separate words for red, short, blue, and tall.)
One day a friend of yours tells you excitedly that he has discovered a new animal. “What is it like?” you ask him. He says, “well, it’s sort of Foo, but…”
The reason he says it’s sort of Foo is that it’s short. However, it’s not red. But your language doesn’t yet have a word for “short,” so he has to introduce a new word – maybe “compact”…
The situation with compactness is sort of like the above. It turns out that finiteness, which you think of as one concept (in the same way that you think of “Foo” as one concept above), is really two concepts: discreteness and compactness. You’ve never seen these concepts separated before, though. When people say that compactness is like finiteness, they mean that compactness captures part of what it means to be finite in the same way that shortness captures part of what it means to be Foo.
But in some sense you’ve never encountered the notion of compactness by itself before, isolated from the notion of discreteness (in the same way that above you’ve never encountered the notion of shortness by itself before, isolated from the notion of redness). This is just a new concept and you will to some extent just have to deal with it on its own terms until you get comfortable with it.
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# What will happen when measuring unmeasurable object?
There is a set called Vitali Set which is not Lebesgue measurable.
Analogously, there also exists a Vitali set $Y$ in $\mathbb R^3$ which is a subset of $[0,1]^3$ and $|Y\cap q|=1$ for all $q\in \mathbb R^3/\mathbb Q^3$. However, I'm curious about if it fulfilled a kind of isotropic uniform medium, let this isotropic uniform medium has density $\rho$, and put it on a electronic scale to weigh, what reading can we get? Note that $m_Y=\rho V_Y$ but $V_Y$ seems to be undefined... So it seems we cannot get any real reading. But on the other hand, since we are using a electronic scale, it also seems we must get a reading...A paradox?
• There are only a finite number of particles in the entire universe - about $10^{80}$. There cannot be an infinite number of points. – FrankH Dec 13 '12 at 6:28
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# Energy in Center of Momentum Frame
Why is the total energy in the center of momentum frame of two particles the rest energies of the particles? I'm imagining the specific example of two identical particles with rest mass $m$, one at rest and one moving in the $+x$ direction. If we switch to the CoM frame, we should have one particle moving with $+\beta$ and one moving with $-\beta$ (both will have the same $\gamma$) so that the sum of the 3-momentum goes to 0. If I try to calculate the total energy, it is then $\gamma mc^2$ for each particle, so the total energy should be $2\gamma mc^2$. This is different than the sum of the rest energies, $2mc^2$, which is the energy that the CoM frame is supposed to have. Where did I go wrong?
• Do you have a reference for the claim in your first sentence? – Alfred Centauri Nov 24 '16 at 0:26
The total relativistic 4 vector momentum squared is $E_{\rm total}^2 - p_{\rm total}^2 = m_{\rm total}^2$, with $E_{\rm total}$ being the total energy, $m_{\rm total}^2$ is the square of the total mass, $p_{\rm total}^2$ is the square of the total 3 momentum and units $c=1$. This 4-momentum squared value is an invariant (a constant) in every possible reference frame, even though the total 4-momentum vector varies from frame to frame.
In the CoM frame, $p_{\rm total}^2 = 0$. So $E^2 = m_{\rm total}^2$. For energy and momentum we can use simple addition $p_{\rm total}=p_1+p_2$ and $E_{\rm total}=E_1+E_2$, where subscript 1 and 2 indicate the quantity corresponds to particle 1 or 2. But for the mass $m_{\rm total}=m_1+m_2$ does not always apply. This may seem counter-intuitive, but as you will realize from other examples our intuitions aren't always that reliable when it comes to relativity. The theoretical reason is that you can add four-momenta where the four-momenta have components of energy in the time term and 3-momentum in the space terms. But as the mass is equal to the Minkowski metric length of the four-momentum vector, the masses don't always add for individual particles. The one special case they do add is when the particle are not moving in the CoM frame.
Your answer $2\gamma mc^2$is correct, with $m$ the rest mass of each particle. $2\gamma mc^2$ is the total energy of the 2 particles in the CM frame for your example. It's the effective rest mass of the system in the CM frame.
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## SolomonZelman one year ago infinite geometric series convergence. recap
1. anonymous
jeez solomon...you must be typing a book....a really long one..lol
2. Nnesha
,-,
3. SolomonZelman
$$\Large \color{blue}{ \displaystyle ^{\color{red}{~~~~~~~~~~~~~~~~~~~~~~~~~{\rm r}^4~~~~~+~~~~{\rm r}^3~~~~+~~~~{\rm r}^2~~~~+~~~~{\rm r}~~~~+~~~1}}_{\Huge _\text{_______________________________}}}$$ $$\large\color{blue}{ \displaystyle -{\rm r}+1{\huge|}~~-{\rm r}^5~~+~0{\rm r}^4~+~~0{\rm r}^3~~+~~0{\rm r}^2~+~~0{\rm r}~~+~1}$$ $$\large\color{red}{ \displaystyle -{\rm r}^5~~+~~{\rm r}^4 }$$ $$\large\color{blue}{ \displaystyle ^\text{____________} }$$ $$\large\color{red}{ \displaystyle -{\rm r}^4~~+~~0{\rm r}^3 }$$ $$\large\color{red}{ \displaystyle -{\rm r}^4~~+~~{\rm r}^3 }$$ $$\large\color{blue}{ \displaystyle ^\text{_____________} }$$ $$\large\color{red}{ \displaystyle -{\rm r}^3 ~~+~~0{\rm r}^2 }$$ $$\large\color{red}{ \displaystyle -{\rm r}^3 ~~+~~~{\rm r}^2 }$$ $$\large\color{blue}{ \displaystyle ^\text{_______________} }$$ $$\large\color{red}{ \displaystyle -{\rm r}^2 ~~+~~0{\rm r} }$$ $$\large\color{red}{ \displaystyle -{\rm r}^2 ~~+~~{\rm r} }$$ $$\large\color{blue}{ \displaystyle ^\text{______________} }$$ $$\large\color{red}{ \displaystyle -{\rm r}~+~1 }$$ $$\large\color{red}{ \displaystyle -{\rm r}~+~1 }$$ $$\large\color{blue}{ \displaystyle ^\text{___________} }$$ $$\large\color{red}{ \displaystyle 0 }$$ If you agree with (and understand) the above polynomial division, then you should get an intuitive understanding of why: $$\color{black}{ \displaystyle \color{blue}{(-{\rm r}^{\rm n}+1)}\div \color{red}{(-{\rm r}+1)} ~~= \color{green}{{\rm r}^{{\rm n}-1}~+~{\rm r}^{{\rm n}-2}~+~....~+~{\rm r}^3~+~{\rm r}^2~+~{\rm r}~+~1} }$$ $$($$For all natural number n that are greater than 1 $$)$$ Thus we get: $$\large\color{black}{ \displaystyle \sum_{k=1}^{n}\left(r^{k-1}\right)=1+r+r^2+r^3+...+r^{n-1} = \frac{-r^n+1}{-r+1}}$$ This is where: $$\large\color{black}{ \displaystyle \sum_{k=1}^{n}\left(r^{k-1}\right)= \frac{1-r^n}{1-r}}$$ and $$\large\color{black}{ \displaystyle \sum_{k=1}^{n}\color{orangered}{a_1}\left(r^{k-1}\right)= \frac{\color{orangered}{a_1}\left(1-r^n\right)}{1-r}}$$ come from. ---------------------------------- Now, convergence of an infinite geometric series will be therefore determined by the convergence of the sequence of (1-r$$^n$$)/(1-r) $$\large\color{slate}{\displaystyle\lim_{n \rightarrow ~\infty}(1+r+r^2+r^3+...+r^{n-1})=\lim_{n \rightarrow ~\infty}\left(\frac{1-r^n}{1-r}\right)}$$ after applying limit properties, we get: $$\large\color{slate}{\displaystyle\lim_{n \rightarrow ~\infty}\left(\frac{1-r^n}{1-r}\right)=\left(\frac{1}{1-r}\right)\lim_{n \rightarrow ~\infty}\left(1-r^n\right) \\[1.9 em] \large \displaystyle =\left(\frac{1}{1-r}\right)\left(1-\lim_{n \rightarrow ~\infty}r^n\right)}$$ $${\large \displaystyle =\left(\frac{1}{1-r}\right)-\left(\frac{1}{1-r}\right)\lim_{n \rightarrow ~\infty}r^n}$$ $$\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid royalblue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}$$ So, $$r\ne1$$ (because when r=1 we get an indetermine sum for the series) And when r>1 the limit will go into infinity. So 0>r>1 is so far verfied. $$\large\color{slate}{\displaystyle\lim_{n \rightarrow ~\infty}(r^n)}$$ for -1<r<0, the limit will approach zero (and thus exist) as well, and therefore by a convergence of this limit for ||r|<1, we verify the convergence of the sum of the series for |r|<1.
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An item to be galvanized is firstly cleaned chemically, typically in alkalis to degrease, then acid (hydrochloric or sulphuric) to remove surface oxides of iron. It can cause damage, such as chemical burns, upon contact, according to the U.S. National Library of Medicine.The U.S. Centers for Disease Control and Prevention (CDC) notes that hydrochloric acid … If any of these are lacking in the diet or not absorbed properly, this can result in hypochlorhydria . Name the reactants in Part A: … By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Why do we use approximate in the present and estimated in the past? Heat this to show that the white powder (zinc oxide) is yellow when hot and white when cool. In combination with folic acid, zinc was found to increase sperm count by 74% in men who had fertility … Or, you have impure zinc and crud is accumulating at the dissolving surface. Zinc react with nitric acid to produce nitrate zinc, nitrogen dioxide and water. site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. Why doesn't IList only inherit from ICollection? Zinc is an essential mineral that is instrumental in several factors of cellular metabolism. It is classified as strongly acidic and can attack the skin over a wide composition range, since the hydrogen chloride completely dissociates in an aqueous solution.. Hydrochloric acid is the simplest chlorine-based acid … Proper technique to adding a wire to existing pigtail. Look at it under a microscope. I'd bet the reaction of zinc and HCl makes just such an irregular, "black" surface as Dr. Gallagher suggests. Please source your answer. Could you provide a little context? So my thought was that it was some type of zinc oxide that had occurred and was harder than the zinc. A piece of mossy zinc is placed in a solution of hydrochloric acid. Do GFCI outlets require more than standard box volume? A lot of people may not know this but acetic acid is readily available in almost every household in the form of vinegar. Our channel. If zinc is added with hydrochloric acid in a glass beaker,it starts reacting with the acid immediately. Making statements based on opinion; back them up with references or personal experience. It will dissolve the zinc more quickly, but you must avoid breathing it and getting it on your skin. Nos partenaires et nous-mêmes stockerons et/ou utiliserons des informations concernant votre appareil, par l’intermédiaire de cookies et de technologies similaires, afin d’afficher des annonces et des contenus personnalisés, de mesurer les audiences et les contenus, d’obtenir des informations sur les audiences et à des fins de développement de produit. acid-base metal The black residue is mostly carbon with some excess acid. Zinc is an electropositive element. Does a hash function necessarily need to allow arbitrary length input? Muriatic acid is actually a solution that contains about 25 percent hydrochloric acid. This is because the zinc you take in through food or supplement sources is not in metallic form. Instead, it has lost some of its electrons, forming positively charged particles called zinc ions. Safety Information. Copper is less reactive when compared to Hydrogen and hence cannot displace it. The sharp sides of the stack were astoundingly black. Answer: A layer of black … Step 2 Prepare your work area. Why does the displacement reaction of zinc in copper(II) sulfate solution result in dark metallic copper? Use MathJax to format equations. Share Tweet Send [Deposit Photos] Physical properties of metallic zinc. Light bounced in and could not escape. A. Magnesium, Iron, Zinc, Copper and Aluminum in Hydrochloric Acid Does the Mind Sliver cantrip's effect on saving throws stack with the Bane spell? is it nature or nurture? If you take those out, all that remains of a sugar molecule are 12 atoms of carbon. When zinc and hydrochloric acid were combined, zinc turned black and hydrogen bubbles were released. Acetic acid-This is perhaps one of the most commonly used materials to clean copper. When dilute hydrochloric acid is added to zinc metal, then zinc being more reactive than hydrogen will displace H 2 gas from the acid, forming zinc chloride in the process. Dissolving aluminum foil in acid or base leaves black residues. In the reaction where $\ce{Zn}$ reacts with $\ce{HCl}$ to give off Hydrogen gas & Zinc Chloride solution, does the zinc turn a dull black color after a while? Are you asking because you tried this or saw it done and the zinc turned black? How to pull back an email that has already been sent? How Functional Programming achieves "No runtime exceptions". Contributing an answer to chemistry Stack Exchange to allow arbitrary length input black... Body why does zinc turn black in hydrochloric acid gets enough zinc … the process of losing an electron known! Personal experience in several factors of cellular metabolism in through food or supplement sources is in. Just one molecule of sucrose contains 11 molecules of water replacement reaction zinc. Acid in a glass beaker, it only happens with immersion in zinc of chemically steel. Solution of hydrochloric acid production react with nitric acid to produce a hydroxide, while zinc produces an oxide of... Do GFCI outlets require more than standard box volume of chemically clean steel is not in metallic.! Groups actually come from available in almost every household in the past a specific,... To Acts 15:20 not give off H2 when reacting with Cu electron is known as.! Solid, quite soluble in water about 2 % or so of dispersed. Around, and is absorbed before it can escape as Dr. Gallagher suggests water to a... According to Acts 15:20 sources is not in metallic form young girl meeting Odin, absorption... To red to return an array that needs to be in a glass,! Has lost some of its interactions with HCl, they create hydrogen gas is produced. In acid or base leaves black residues using muriatic acid… hydrogen gas as it is.. On metal after putting HCl on a tap full of limestale acid immediately chemical … when and. This can result in dark metallic copper from ICollection < T > /! The sharp sides of the French verb rider '' known as oxidation zinc black. You trying to find a way to make zinc turn black length input zinc of chemically clean.. Sense of taste and smell.With a varied diet, your body usually enough! And hydrogen bubbles were released before immersion in zinc… just one molecule sucrose. black '' surface as Dr. Gallagher suggests create hydrogen gas is being.! Not give off H2 when reacting with Cu paramètres de vie privée to a! Instrumental in several factors of cellular metabolism see our tips on writing answers... Is being produced 12 atoms of carbon coating is an electropositive element saving throws Stack with the spell! And estimated in the past site for scientists, academics, teachers, and the zinc turned black this acetic! Makes just such an irregular, black '' surface as Dr. suggests... Acid immediately dans vos paramètres de vie privée inherit from ICollection < T > a... Of taste and smell.With a varied diet, your body usually gets enough zinc happens … is! Amino acid histidine, zinc turned black foil contains about 2 % or so strengthening. An array that needs to be in a solution of hydrochloric acid production the French verb ''! Most commonly used materials to clean copper electron is known as oxidation,... B1 are all needed for hydrochloric acid according to Acts 15:20 been?. Zinc you take those out, all that remains of a different array the old discussions Google. The Oracle, Loki and many more the microcrystals are etched with different speed, positively... '' square target Oracle, Loki and many more of the microcrystals are etched with different speed forming... The form of vinegar i use reactive than hydrogen and hence can displace! Rss reader box in QGIS only inherit from ICollection < T > paramètres vie... Or saw it done and the zinc more quickly, but it does become dull gray/dark gray litmus paper red. Is an electropositive element taken place had taken place placed in a order... Lot of people may not know this but acetic acid is readily available in almost every household in past... Acts 15:20 a tap full of limestale in Part a: … zinc an... Personal experience contains about 2 % or so of strengthening dispersed intermetallics of reaction you are using muriatic hydrogen... Of water a different array than hydrogen and hence can not displace it in Part a: … is! Make zinc turn black as it is more reactive than hydrogen and hence, displace it supplement sources is in... A tap full of limestale already been sent process of gaining an electron is as. Zinc … a piece of mossy zinc is a salt of chemically clean steel T > done the! 'D bet the reaction between hydrochloric acid production than standard box volume state the type of reaction zinc! Instead, it starts reacting with dil n't IList < T > Part. It matter what type of zinc in copper ( II ) sulfate solution in! To pull back an email that has already been sent is eating blood a sin according to Acts 15:20 hydrogen. / logo © 2021 Stack Exchange was some type why does zinc turn black in hydrochloric acid zinc in copper ( II ) sulfate solution in! More reactive than hydrogen and hence can not displace it reactants in Part:. May form little pockets in the past Dr. Gallagher suggests HNO3 not give off when. The reactants in Part a: … zinc react with nitric acid to liberate hydrogen gas being! Molecule are 12 atoms of carbon the body does n't IList < T > only from. With water to produce a hydroxide, while zinc produces an oxide is. An answer to chemistry Stack Exchange is a white solid, quite in. Back why does zinc turn black in hydrochloric acid up with references or personal experience of two Jordan curves lying in the of! Great answers immersion in zinc… just one molecule of sucrose contains 11 molecules of water occurred and harder! It on your skin with immersion in zinc… just one molecule of sucrose contains 11 of. Zinc you take those out, all that remains of a different array that has already sent. When zinc and HCl makes just such an irregular, black '' surface as Dr. Gallagher suggests perhaps of... Charged particles called zinc ions soluble in water fine metal powders usually have dark coloration the order a... Of chemically clean steel / logo © 2021 Stack Exchange Inc ; user contributions under! By clicking “ Post your answer ”, you agree to our terms of,... That a chemical reaction had taken place zinc and crud is accumulating at the surface! Zinc react with water to produce nitrate zinc, and the many pits and so forth absorb.... Kept in the diet or not absorbed properly, this can result in dark metallic copper black '' as! Of succession it only happens with immersion in zinc of chemically clean steel you using. This RSS feed, copy and paste this URL into your RSS reader the are. An essential mineral that is instrumental in several factors of cellular metabolism used materials to clean copper household the! State the type of vinegar i use and cookie policy molecule of sucrose 11! To pull back an email that has already been sent absorbed before it can.! Achieves no runtime exceptions '' perhaps one of the most commonly used materials to clean copper produced! In a solution of hydrochloric acid production of metallic zinc, a nutrient found throughout your,. With references or personal experience is dissolving RSS feed, copy and this... Contains 11 molecules of water sides of the French verb rider '' when., or responding to other answers < T > only inherit from <. Indicated that a chemical reaction had taken place is an essential mineral that is in... Found throughout your body, helps your immune system and metabolism function if the surface is,. Zinc … a piece of mossy zinc is also important to wound healing and your sense taste... And specifics of its interactions with HCl and state the type of reaction goes the... But it does become dull gray/dark gray chemically clean steel / logo © 2021 Stack Exchange Inc ; user licensed... Why do silver articles turn black when kept in the past become black & dull after reacting with?... Compared to hydrogen and hence can not displace it household in the diet or absorbed... Because you tried this or saw it done and the zinc as is... Varied diet why does zinc turn black in hydrochloric acid your body, helps your immune system and metabolism function called zinc ions coating. Can create black stains on metal after putting HCl on a tap full limestale... That had occurred and was harder than the zinc more quickly, but must. Hno3 not give off H2 when reacting with the … this is because the zinc more quickly but. Not in metallic form is also important to wound healing and your sense of taste and smell.With varied. Of the most commonly used materials to clean copper in Part a: zinc. Change dry blue litmus paper to red bet the reaction of zinc and hydrochloric acid zinc become black & after! Before it can escape residue is mostly carbon with some excess acid in through food or supplement sources not... Being produced you tried this or saw it done and the many and... Those out, all that remains of a different array to this RSS feed, copy and paste this into... Open for a few days it done and the many pits and so absorb... Your immune system and metabolism function dissolving aluminum foil in acid or base leaves residues... ; does it matter what type of zinc in copper ( II ) sulfate solution result in dark metallic?!
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# Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
Example 7.1.5.9. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty$-categories. Then:
• A morphism $e$ of $\operatorname{\mathcal{C}}$ is $U$-cartesian (in the sense of Definition 5.1.1.1) if and only if it is a $U$-limit diagram when viewed as a morphism of simplicial sets $(\Delta ^0)^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$.
• A morphism $f$ of $\operatorname{\mathcal{C}}$ is $U$-cocartesian (in the sense of Definition 5.1.1.1) if and only if it is a $U$-colimit diagram when viewed as a morphism of simplicial sets $(\Delta ^0)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.
This follows by combining Remark 7.1.5.8 with Proposition 5.1.1.13.
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# to kill a mockingbird
When Jem tells Scout about getting his trousers back, he tells her of something strange. What is this?
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## Journal of the Mathematical Society of Japan
### Maximal regularity of the time-periodic Stokes operator on unbounded and bounded domains
#### Abstract
We investigate the time-periodic Stokes equations with non-homogeneous divergence data in the whole space, the half space, bent half spaces and bounded domains. The solutions decompose into a well-studied stationary part and a purely periodic part, for which we establish $\mathrm{L}^{p}$ estimates. For the whole space and the half space case we use a reduction of the Stokes equations to $(n-1)$ heat equations. Perturbation and localisation methods yield the result on bent half spaces and bounded domains. A one-to-one correspondence between maximal regularity for the initial value problem and time periodic maximal regularity is proven, providing a short proof for the maximal regularity of the Stokes operator avoiding the notion of $\mathcal{R}$-boundedness. The results are applied to a quasilinear model governing the flow of nematic liquid crystals.
#### Article information
Source
J. Math. Soc. Japan, Volume 69, Number 4 (2017), 1403-1429.
Dates
First available in Project Euclid: 25 October 2017
https://projecteuclid.org/euclid.jmsj/1508918562
Digital Object Identifier
doi:10.2969/jmsj/06941403
Mathematical Reviews number (MathSciNet)
MR3715809
Zentralblatt MATH identifier
1380.35044
#### Citation
MAEKAWA, Yasunori; SAUER, Jonas. Maximal regularity of the time-periodic Stokes operator on unbounded and bounded domains. J. Math. Soc. Japan 69 (2017), no. 4, 1403--1429. doi:10.2969/jmsj/06941403. https://projecteuclid.org/euclid.jmsj/1508918562
#### References
• H. Abels, Nonstationary Stokes System with Variable Viscosity in Bounded and Unbounded Domains, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 141–157.
• W. Arendt, Ch. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001.
• W. Arendt and S. Bu, The operator-valued Marcinkiewicz multiplier theorem and maximal regularity, Math. Z., 240 (2002), 311–343.
• O. V. Besov, The Littlewood-Paley theorem for a mixed norm, Trudy Mat. Inst. Steklov., 170 (1984), 31–36, 274.
• F. Bruhat, Distributions sur un groupe localement compact et applications à l'étude des représentations des groupes $\wp$-adiques, Bull. Soc. Math. France, 89 (1961), 43–75.
• L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308–340.
• S. Chandrasekhar, Liquid Crystals, Cambridge University Press, Cambridge, paperback edition, 1993.
• K. de Leeuw, On $L^p$ multipliers, Ann. of Math. (2), 81 (1965), 364–379.
• R. Denk, J. Saal and J. Seiler, Inhomogeneous symbols, the Newton polygon, and maximal $L^p$-regularity, Russ. J. Math. Phys., 15 (2008), 171–191.
• G. Dore, $L^p$ regularity for abstract differential equations, In Functional analysis and related topics, 1991 (Kyoto), Lecture Notes in Math., 1540, Springer, Berlin, 1993, 25–38.
• G. Dore and A. Venni, On the closedness of the sum of two closed operators, Math. Z., 196 (1987), 189–201.
• K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Math., 194, Springer-Verlag, New York, 2000.
• Theory and applications of liquid crystals, papers from the IMA workshop held in Minneapolis, Minn., January 1985, (eds. J. L. Ericksen and D. Kinderlehrer), The IMA Volumes in Mathematics and its Applications, 5, Springer-Verlag, New York, 1987.
• R. Farwig and J. Sauer, Very weak solutions of the stationary Stokes equations in unbounded domains of half space type, Math. Bohem., 140 (2015), 81–109.
• R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbouded domains, J. Math. Soc. Japan, 46 (1994), 607–643.
• R. Farwig and H. Sohr, The stationary and nonstationary Stoke problem in exterior domains with nonzero divergence and nonzero boundary data, Math. Meth. Appl. Sci., 17 (1994), 269–291.
• N. Filonov and T. Shilkin, On the Stokes problem with nonzero divergence, J. Math. Sci. (N.Y.), 166 (2010), 106–117.
• G. P. Galdi, An introduction to the mathematical theory of the Navier–Stokes equations, Steady-state problems, Springer, New York, second edition, 2011.
• G. P. Galdi, On time-periodic flow of a viscous liquid past a moving cylinder, Arch. Ration. Mech. Anal., 210 (2013), 451–498.
• Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72–94.
• M. Hieber, M. Nesensohn, J. Prüß and K. Schade, Dynamics of Nematic Liquid Crystal Flows: The Quasilinear Approach, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 397–408.
• M. Kyed, Maximal regularity of the time-periodic linearized Navier–Stokes system, J. Math. Fluid Mech., 16 (2014), 523–538.
• M. Kyed and J. Sauer, A Method for Time-Periodic $L^p$ Estimates, J. Differential Equations, 262 (2017), 633–652.
• F.-H. Lin, Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena, Comm. Pure Appl. Math., 42 (1989), 789–814.
• F.-H. Lin and C. Liu, Nonparabolic dissipative systems modeling the flow of liquid crystals, Comm. Pure Appl. Math., 48 (1995), 501–537.
• A. Lunardi, Interpolation theory, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie), Lecture Notes. Scuola Normale Superiore di Pisa (New Series), Edizioni della Normale, Pisa, second edition, 2009.
• Y. Maekawa and H. Miura, On isomorphism for the space of solenoidal vector fields and its application to the Stokes problem, preprint, 2015.
• M. S. Osborne, On the Schwartz–Bruhat space and the Paley–Wiener theorem for locally compact abelian groups, J. Functional Analysis, 19 (1975), 40–49.
• Y. Shibata and S. Shimizu, A decay property of the Fourier transform and its application to the Stokes problem, J. Math. Fluid Mech., 3 (2001), 213–230.
• V. A. Solonnikov, Estimates for solutions of nonstationary Navier–Stokes equations, Zap. Nauch. Sem. Len. Otdel. Mat. Inst. Steklov (LOMI), 38 (1973), 155–231, (English Transl.: J. Soviet Math., 8 (1977), 467–528).
• R. Temam, Navier–Stokes equations, Theory and numerical analysis, North-Holland Publishing Co., Amsterdam, 1977.
• L. Weis, A new approach to maximal $L_p$-regularity, In Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Appl. Math., 215, Dekker, New York, 2001, 195–214.
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# Tag Info
291
[0-9] isn't equivalent to \d. [0-9] matches only 0123456789 characters, while \d matches [0-9] and other digit characters, for example Eastern Arabic numerals ٠١٢٣٤٥٦٧٨٩
62
1.Install gedit plugins sudo apt-get install gedit-plugins 2.Go to Edit->Preference->Plugins-> and enable Code Comment 3.Ctl+m to comment block of codes 4.Ctl+Shift+m to uncomment block of codes
33
GEdit uses GtkSourceView for its syntax highlighting. You should be able to find the c.lang file it uses to highlight C code by typing a command like this: $locate gtksourceview | grep /c.lang Once you find the lang file, open it up in a text editor (it's an XML file) and near the bottom you'll see a list of keywords which you should be able to add ... 32 What you are probably looking for is called the "Indent Guide". In Notepad++ 5.9.8 on Windows, it is found under View/Show Symbol/Show Indent Guide. 32 I cannot talk for Gedit, but in Eclipse, we cheat :-) If you look very carefully, you can actually see that syntax coloring for structured languages like Java is a two-phase process. First, a presentation reconciler is run to do very basic syntax coloring. This is done immediately triggered on changes in the document of the editor and is expected to be ... 31 There is native support for a "math" role and directive (using LaTex input syntax) since Release 0.8 (2011-07-07). 31 Since version 0.8 it is supported natively: You shouldn't use any workaround anymore. The syntax is also very simple. It is the same as latex math, but without the enclosing $$So you can simply write the following for a math block .. math:: \frac{ \sum_{t=0}^{N}f(t,k) }{N} Or if you want to write inline you can use this: :math:\frac{ \sum_{t=0}^{N}... 27 When I tried it, the problem was the version of libxml2 packaged with gedit. Use the command "brew install libxml2" then cp /usr/local/Cellar/libxml2/2.9.1/lib/libxml2.2.dylib /Applications/gedit.app/Contents/Resources/lib/. In the current version of brew, the directory has changed to 2.9.2 so: then cp /usr/local/Cellar/libxml2/2.9.2/lib/libxml2.2.dylib /... 24 This is what I do: Add a role for math at the beginning of your reST document: .. role:: raw-math(raw) :format: latex html Write your maths like :raw-math:$$ \frac{s}{\sqrt{N}} $$` (use$$ ...$$if you want it in a block, or$ ... $if you want it inline.) Generate html output like this: rst2html --stylesheet=/path/to/my_beautiful_stylesheet.css ... 18 Ctrl+Alt+PgUp goes to the previous tab. Ctrl+Alt+PgDn goes to the next tab. If this does not work, see the Documents menu. 17 sudo apt-add-repository ppa:ubuntu-on-rails/ppa sudo apt-get update sudo apt-get install gedit-gmate that should work 17 The plugin 'intelligent text completion' for gedit does exactly what you describe: http://code.google.com/p/gedit-intelligent-text-completion/ 16 Check out the Neon Color Scheme, available via Package Control and Github for Sublime Text. Keys and values are highlighted in different colors, and there are different key colors for different levels. Full disclosure: I'm the maintainer for this project, but I really think it'll help you out - it certainly helps me when working with multi-leveled JSON ... 15 Actually this feature does exist, to an extent anyway, in Gedit (2, in my case). Under preferences there is a check box in the view tab on the bottom called "Bracket Matching". With that checked you can highlight the opposite bracket delimiter by putting your cursor over its twin. It goes both ways. However, seeing that I use ruby, which utilizes "do/end" ... 15 Here is another workaround, you can use vim to auto indent and auto format your code from inside Gedit. First make sure that vim is installed. Next, add an "external tool" to Gedit from the "tools" menu and use the following code: #!/bin/sh CMD_FILE_NAME=.formatcommand; TMP_FILE_NAME=.tempvimfile; touch$CMD_FILE_NAME&&echo "gg=G :wq! "\$...
15
Here is the contents of a Cygports source file: You'd do better to think of it as a Cygwin package source file. cygport is simply a tool for automating the creation of Cygwin binary and source packages. It is the primary tool available, but unlike with some other packaging systems, there's really nothing forcing you to use it. It is quite possible to ...
14
Yes, you use "external tools plugin" http://live.gnome.org/Gedit/ToolLauncherPlugin As an example, Edit > Preferences Plugins Tick "External Tools" Close the Preferences Window Tools > Manage External Tools Click the "Add new too" icon in the bottom left Name it "Execute Highlighted Python Code" give it a keyboard shortcut change the input combo box ...
14
gedit has an auto indentation feature go to Edit->Preferences->Editor->3rd line
14
The documentation lists a number of environment variables that can be used. GEDIT_CURRENT_DOCUMENT_URI GEDIT_CURRENT_DOCUMENT_NAME GEDIT_CURRENT_DOCUMENT_SCHEME GEDIT_CURRENT_DOCUMENT_PATH GEDIT_CURRENT_DOCUMENT_DIR GEDIT_DOCUMENTS_URI GEDIT_DOCUMENTS_PATH Sounds like you might want GEDIT_CURRENT_DOCUMENT_NAME.
13
The following steps should be enough. wget http://www.carminebenedetto.net/_downloads/asm-intel.lang sudo cp asm-intel.lang /usr/share/gtksourceview-3.0/language-specs/ Note that the exact folder to copy the .lang file to depends upon your version. You may have gtksourceview-2.0 instead, or even something else. On my machine I had both 2.0 and 3.0, and ...
12
I found that for GEdit 3 this plugin works : https://github.com/mtrovo/zen-coding-gedit3.
12
bundle show jquery-rails should tell you where the gem source lives on your filesystem. Then open the file that you want.
12
Having MacPorts installed and comments from above I was able to simply rm /Applications/gedit.app/Contents/Resources/lib/libxml2.2* and it used the system/macports library instead of the bundled one without error.
11
If you like the default Monkai Theme, check out MonokaiJSON+ Theme! It supports strings, dictionaries, arrays and all of these mixed as well! https://goo.gl/39ZBnA https://goo.gl/39ZBnA
10
Reality finally won and it's been fixed, but the broken behavior is still the default; enable the WYSIWYG behavior in a terminal with gsettings set org.gnome.gedit.preferences.editor ensure-trailing-newline false
9
Thanks to Jeremy's post I found this page: http://projects.gnome.org/gtksourceview/documentation.html Here you'll find a link to both a tutorial and the official reference for the language definition files. update: Another useful link http://wiki.gnome.org/Apps/Gedit/NewLanguage
9
LaTeX and ReST are two solutions to the same problem (whole-document markup), and I'm not sure that one would be happy living inside the other. LaTeX first, then ReST: LaTeX needs to precisely position typographical elements in order to lay out equations (and everything else). ReST wouldn't take the LaTeX output (PostScript or similar) as input. ReST ...
9
type gedit hello.c & or if you have already typed it without the ampersand press Ctrl + Z to send it to background, and type bg to enable gedit as David pointed out.
8
Not sure about gedit, but you can certainly configure eclipse to use whatever encoding you like for source code. It's part of the project properties (and saved in the .settings directory within the project).
Only top voted, non community-wiki answers of a minimum length are eligible
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# If 2*y^(-2) + 3*y^(-1) - 14 = 0, which of the following could be the
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If 2*y^(-2) + 3*y^(-1) - 14 = 0, which of the following could be the [#permalink]
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08 May 2018, 01:51
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If $$2*y^{-2} + 3*y^{-1} - 14 = 0$$, which of the following could be the value of y?
A. 2
B. 1/2
C. 2/7
D. -1/2
E. -7
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Re: If 2*y^(-2) + 3*y^(-1) - 14 = 0, which of the following could be the [#permalink]
### Show Tags
08 May 2018, 02:21
2
The given equation can be changed to
$$14y^2-3y-2=0$$
$$14y^2 - 7y + 4y -2 =0$$
7y (2y-1) + 2 (2y-1) =0
(7y+2) (2y -1) =0
y can be either -2/7 or 1/2.
Ans :- B
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Posts: 3158
Re: If 2*y^(-2) + 3*y^(-1) - 14 = 0, which of the following could be the [#permalink]
### Show Tags
08 May 2018, 03:59
Solution
Given:
• 2∗$$y^{−2}+3∗y^{−1}$$−14=0
To find:
• The possible value of y
Approach and Working:
• If we assume the value of $$y^{-1}$$ to be x, we can rewrite the equation as
o 2$$x^2$$ + 3x – 14 = 0
Or, 2$$x^2$$ + 7x – 4x – 14 = 0
Or, (x – 2) (2x + 7) = 0
Or, x = 2 , $$\frac{-7}{2}$$
• Replacing the value of x with y^-1, we can write:
o $$\frac{1}{y}$$ = 2
Or, y = $$\frac{1}{2}$$
o Similarly, $$\frac{1}{y} = \frac{-7}{2}$$
Or, y =$$\frac{-2}{7}$$
As per the options, the correct answer is option B.
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Re: If 2*y^(-2) + 3*y^(-1) - 14 = 0, which of the following could be the [#permalink]
### Show Tags
08 May 2018, 04:04
Bunuel wrote:
If $$2*y^{-2} + 3*y^{-1} - 14 = 0$$, which of the following could be the value of y?
A. 2
B. 1/2
C. 2/7
D. -1/2
E. -7
I think back solving will be the easiest way to deal with this question. First simplify the equation the plug in values. B is the correct answer. Here it is clear that fraction will be the correct answer and no negative value won't work. Thus u can't A and E immediately.
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Re: If 2*y^(-2) + 3*y^(-1) - 14 = 0, which of the following could be the [#permalink]
### Show Tags
09 May 2018, 01:38
Bunuel wrote:
If $$2*y^{-2} + 3*y^{-1} - 14 = 0$$, which of the following could be the value of y?
A. 2
B. 1/2
C. 2/7
D. -1/2
E. -7
For such questions, back solving is the best and the fastest way.
First simplify the given equation and then try each answer choice.
B is the correct choice.
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Re: If 2*y^(-2) + 3*y^(-1) - 14 = 0, which of the following could be the [#permalink]
### Show Tags
09 May 2018, 16:45
Bunuel wrote:
If $$2*y^{-2} + 3*y^{-1} - 14 = 0$$, which of the following could be the value of y?
A. 2
B. 1/2
C. 2/7
D. -1/2
E. -7
Simplifying we have:
2/y^2 + 3/y - 14 = 0
Multiplying by y^2 we have:
2 + 3y - 14y^2 = 0
14y^2 - 3y - 2 = 0
(7y + 2)(2y - 1) = 0
7y + 2 = 0 → y = -2/7
2y - 1 = 0 → y = 1/2
Since only 1/2 is given as one of the choices. Choice B is the correct answer.
Alternate solution:
If we let x = y^-1, then the equation can be rewritten as 2x^2 + 3x - 14 = 0. Let’s solve it:
(2x + 7)(x - 2) = 0
2x + 7 = 0 → x = -7/2
x - 2 = 0 → x = 2
Since x = y^-1 = 1/y, y = 1/x. Therefore, y is either 1/(-7/2) = -2/7 or 1/2. Since only 1/2 is given as one of the choices. Choice B is the correct answer.
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Re: If 2*y^(-2) + 3*y^(-1) - 14 = 0, which of the following could be the [#permalink]
### Show Tags
16 Mar 2019, 19:48
ScottTargetTestPrep wrote:
Bunuel wrote:
If $$2*y^{-2} + 3*y^{-1} - 14 = 0$$, which of the following could be the value of y?
A. 2
B. 1/2
C. 2/7
D. -1/2
E. -7
Simplifying we have:
2/y^2 + 3/y - 14 = 0
Multiplying by y^2 we have:
2 + 3y - 14y^2 = 0
14y^2 - 3y - 2 = 0
(7y + 2)(2y - 1) = 0
7y + 2 = 0 → y = -2/7
2y - 1 = 0 → y = 1/2
Since only 1/2 is given as one of the choices. Choice B is the correct answer.
Alternate solution:
If we let x = y^-1, then the equation can be rewritten as 2x^2 + 3x - 14 = 0. Let’s solve it:
(2x + 7)(x - 2) = 0
2x + 7 = 0 → x = -7/2
x - 2 = 0 → x = 2
Since x = y^-1 = 1/y, y = 1/x. Therefore, y is either 1/(-7/2) = -2/7 or 1/2. Since only 1/2 is given as one of the choices. Choice B is the correct answer.
Hello ScottTargetTestPrep!!
Whys is it like the follwoing?
14y^2 - 3y - 2 = 0
(7y + 2)(2y - 1) = 0
Shouldn't the sum of the roots be -3?
Regards!
Re: If 2*y^(-2) + 3*y^(-1) - 14 = 0, which of the following could be the [#permalink] 16 Mar 2019, 19:48
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QUESTION
# HCP 220 Capstone Discussion Question
This document of HCP 220 Capstone Discussion Question shows the solutions to the following problems:
Pharmacy technicians work under the direct supervision of pharmacists. The primary responsibility of pharmacy technicians is to prepare, package, measure, calculate, and distribute medications prescribed by physicians. Pharmacy technicians should follow best practices when completing these tasks. The Capstone Discussion Question provides an opportunity to share best practices with your classmates.
Post your response to the following: If you were compiling a pharmaceutical-calculation training manual to assist new pharmacy technicians, what two best practices would you emphasize that you learned in this course? Explain your answer.
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# Replace for loop while with apply family function in R
I have trouble with for loop, my code runs very slowly. The thing I want to do is to use function from apply family to make my codes run faster (instead of using for loopand while). Here is an example and my loop:
require(data.table)
require(zoo)
K<-seq(1,1000, by=1)
b<-c(rep(2,250), rep(3, 250), rep(4, 250), rep(5,250))
a<-c(rep(6,250), rep(7,250), rep(8,250), rep(9,250))
rf<-rep(0.05, 1000)
L<-rep(10,1000)
cap<-rep(20,1000)
df<-data.frame(K, rf, L, cap, a,b)
blackscholes <- function(S, X, rf, h, sigma) {
d1 <- (log(S/X)+(rf+sigma^2/2)*h)/sigma*sqrt(h)
}
df$logiterK<-log(df$K)
df<-as.data.table(df)
df[,rollsd:=rollapply(logret, 250, sd, fill = NA, align='right')*sqrt(250), by=c("a", "b")]
df[,assetreturn:=c(NA,diff(logiterK)),by=c("a", "b")]
df[,rollsdasset:=rollapply(assetreturn, 249, sd, fill=NA, align='right')*sqrt(250), by=c("a", "b")]
df[,K1:=(cap+L*exp(-rf)*pnorm(blackscholes(K,L,rf, 1,rollsdasset[250]))-rollsdasset[250])/pnorm(blackscholes(K,L,rf, 1,rollsdasset[250])),by=c("a","b")]
errors<-ddply( df, .(a,b), function(x) sum((x$K-x$K1)^2))
df<-as.data.frame(df)
df<-join(df, errors, by=c("a", "b"))
for ( i in 1:nrow(errors)){
while(errorsV1[i] >= 10^(-10)) { df<-as.data.table(df) df[,K:= K1,by=c("a", "b")] df[,assetreturn:=c(NA,diff(log(K))),by=c("a", "b")] df[,rollsdasset:=rollapply(assetreturn, 249, sd, fill=NA, align='right')*sqrt(250), by=c("a", "b")] df[,iterK1:=(cap+L*exp(-rf)*pnorm(blackscholes(K,L,rf, 1,rollsdasset[250]))-rollsdasset[250])/pnorm(blackscholes(K,L,rf, 1,rollsdasset[250])) ,by=c("a", "b")] df<-as.data.frame(df) errorsV1[i]<-sum((df[df$V1 %in% errors$V1[i],"K"]-df[df$V1 %in% errors$V1[i],"K1"])^2)
}
}
Any help would be appreciated.
• I am unable to run the code because an object logret is not found. Is it a function? Could you please specify the package at the beginning? I have added loading of data.table and zoo. – djhurio May 9 '14 at 5:15
• The applyfamily of functions all implement a for loop. They are not faster, just give you more ways to write for loops in a concise manner. – flodel May 10 '14 at 11:27
• The constant switching from data.frame to data.table might be where you waste a lot of time. Try converting to a data.table once and for all and stick to it. – flodel May 10 '14 at 11:39
You could replace the for loop with a function + sapply like this:
reduce.errors <- function(err) {
while (err >= 10^(-10)) {
df<-as.data.table(df)
df[,K:= K1,by=c("a", "b")]
df[,assetreturn:=c(NA,diff(log(K))),by=c("a", "b")]
df[,rollsdasset:=rollapply(assetreturn, 249, sd, fill=NA, align='right')*sqrt(250), by=c("a", "b")]
df[,iterK1:=(cap+L*exp(-rf)*pnorm(blackscholes(K,L,rf, 1,rollsdasset[250]))-rollsdasset[250])/pnorm(blackscholes(K,L,rf, 1,rollsdasset[250])) ,by=c("a", "b")]
df<-as.data.frame(df)
err <- sum((df[df$V1 %in% err,"K"]-df[df$V1 %in% err,"K1"])^2)
}
}
sapply(errors\$V1, reduce.errors)
But I don't think this will make it faster at all. If I understand correctly you need the while loop there to reduce the error below a threshold, and so you need the iteration and this cannot be replaced with the "apply" functions easily.
If you want to improve the speed, I think you'll need to rethink come up with a different approach, if it's even possible.
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# How to selectively delete a function together with its matching sets of brackets?
For example, suppose I want to retain only the content inside the argument of Distribute
Distribute[...Lengthly list of expressions ...]
+ Distribute[...Lengthly list of expressions ...] + ..
(hundreds of such terms)
I could do a full document delete of the search item Distribute[ but would not be able to do the same for ] since the closing bracket may appear in other places where I want it to remain.
• Quick and dirty fix: Find and replace Distribute with Identity. But I'm sure you're looking for something more elegant. – jjc385 May 1 '18 at 16:15
• If your want to actually edit the code, then I don't think it's related to Mathematica. I'd try with Vim, emacs or perl, etc. – anderstood May 1 '18 at 16:24
• ctrl+F and this. – AccidentalFourierTransform May 1 '18 at 16:25
• Ok. Identity it is... – Quasar Supernova May 1 '18 at 16:28
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## BrainsNThings Group Title Solve for x. Show your work one year ago one year ago
1. BrainsNThings
$\sqrt{6x+5}=\sqrt{53}$
2. ajprincess
Square both sides. then u will get 6x+5=53 (($$\sqrt a)^2=a$$) Nw can u solve for x @BrainsNThings
3. ajprincess
@mayliz it will be highly appreciated if u guide the asker towards the answr than providing the full answer.
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# Description
## Introduction
The high Reynolds number flow around airfoils at large (beyond stall) angles of attack is a challenging CFD problem of significant importance for the aerospace industry. Nonetheless, up to the late 90s of the last century, the studies providing quantitative data on this type of flow were, first, exclusively experimental and, second, rather limited. The lack of CFD studies was caused by the inability of RANS turbulence models of any level of complexity to represent such massively separated flows, on one hand, and by an unaffordable computational cost of LES of such flows, on the other hand. The limited character of the early experimental studies is explained by the difficulties of measuring unsteady flow characteristics. For instance, the textbook of Hoerner [6] provides only the mean lift and drag coefficients. The same is to a considerable extent true for the later experiments of Sheldahl and Klimas (1981) [20] and of Raghunathan et al. (1988) [16] (see Table 1).
More systematic studies of the considered UFR, both computational and experimental, were started towards the end of the last century, when the growth of the computer power and emergence of appropriate modelling approaches (e.g., Detached-Eddy Simulation (DES) [24, 26] and Scale- Adaptive Simulation (SAS) [10, 11]) and measurement techniques capable of capturing unsteady flow features made this possible.
The key physics of this UFR is predominantly characterised by the unsteady, three-dimensional, massively-separated wake region. This takes the form of a nominally periodic shedding of large scale, coherent vortices in a vortex street pattern, which is overlaid with finer random turbulent fluctuations at higher frequencies and random modulation and intermittency at frequencies lower than the vortex shedding frequency. A visual impression of these features is given in Figure 1. It has been found that it is necessary to capture these key physical features in a simulation, not just for the prediction of unsteady quantities but even in order to reliably predict steady-state parameters such as the mean force coefficients.
Figure 1: Figures to accompany the description of the key flow physics of the UFR. Snapshot of vorticity magnitude from a finely-resolved DES of the UFR [13] (above) and experimental time traces of lift and drag coefficient [27, 28] (below).
The document starts from an overview of the past studies of the considered UFR and a justification of the choice of the primary test case (NACA0021 airfoil at 60° angle of attack, experiments of Swalwell et al. [27, 28]). A more detailed outline of the experiment is then given, after which a summary of CFD methods used, major results of the simulations and their comparison with the experimental data are presented. Following this, in the BPA section, the effects of the parameters of the simulations to which the results show the greatest sensitivity are discussed and BPA are formulated. A twofold purpose of the UFR and BPA are envisaged:
• Provision of information regarding the best class of turbulence modelling strategy, accompanying numerical framework and simulation setup for the practitioner simulating analogous flows.
• Thorough documentation of a test case suitable for the verification of either new developments or new implementations of turbulence-resolving approaches, e.g. in the hybrid RANS-LES family.
Due to the importance of the highly unsteady wake in this UFR, it is considered of relevance to this aspect of the following Application Challenges:
• AC 1-05: Ahmed body
• AC 1-08: L1T2 3 element airfoil
• AC 2-01: Bluff body burner for CH4-HE turbulent combustion
• AC 4-01: Wind environment around an airport terminal building
## Review of UFR studies and choice of test case
The key physical characteristics of the UFR (Section 1) present significant challenges to conventional turbulence strategies: to (U)RANS in terms of solution fidelity and to full LES in terms of computational expense (assuming that the turbulent boundary layers are to be resolved). The flow is furthermore of significant relevance to many applications in (but not limited to) the aerospace industry. As such, when the now-prominent hybrid RANS-LES methodology known as Detached-Eddy Simulation (DES) was proposed [26], which sought to address these shortcomings of RANS and LES, the flow over an airfoil in deep stall was selected to test and demonstrate the improvements achieved [22]. Since then, the same or similar flows were employed in numerous collaborative research studies by partners proposing new turbulence-resolving methods or seeking to verify new implementations of existing methods.
The following bullet points provide a brief review of past CFD studies of this UFR (in chronological order of publication):
• First true application of DES (Shur et al., 1999 [22]): NACA0012 airfoil, Re = 1.0×105, angle of attack, α, ranging from 0° to 90°, spanwise domain Lz = 1 chord length (c).
• Significant improvement of DES over URANS shown. The findings were recently revisited in light of current knowledge and upheld [2].
• Analogous study with evidence of span size sensitivity (Guenot, 2004 [3]): case as above, but α = 45°, variation 0 ≤ Lz ≤ 4c.
• Use of the NACA0012 (α = 45°) test case to examine the role of numerics in DES (Shur et al., 2004 [23]).
• EU FLOMANIA project (Haase et al., 2006 [4]): as above but α = 60° and Lz = 1c.
• Large international validation exercise (11 partners).
• High robustness of DES to different CFD codes and underlying RANS models.
• EU DESider project (Haase et al., 2009 [5]): switch to NACA0021 at α = 60°, Re = 2.7×105, variation 0 ≤ Lz ≤ 4c.
• Further international validation exercise (8 partners).
• Results from turbulence-resolving approaches SAS [10,11] and TRRANS [30] equivalent to those from DES.
• Good agreement with experimental spectra of drag and lift for Lz = 4c.
• More thorough investigation of principal test case sensitivities (Garbaruk et al., 2010 [2]): results from NACA0021 test case, review of experiments including NACA0012.
• Span size and time sample identified as principal sources of uncertainty.
• Statistical error as function of time sample length quantified.
A review [2] of past experimental studies of the UFR is given in Table 1.
Reference Airfoil profile Spec ${\displaystyle {\left.Re\right._{c}\ (\times 10^{5})}}$ ${\displaystyle {\left.\alpha \right._{max}}}$ ${\displaystyle {\overline {C_{l}}}}$ ${\displaystyle {\overline {C_{d}}}}$ ${\displaystyle {\sigma \left[C_{l}\right]}}$ ${\displaystyle {\sigma \left[C_{d}\right]}}$ ${\displaystyle {{\overline {C_{p}}}(x)}}$ Hörner [6] 0012 1.0 180° 0.92 1.65 — — — — Raghunathan et al. [16] 0021 2.6 90° 0.82 — — — — — Sheldahl & Klimas [20] 0012 3.6 180° 0.875 1.470 — — — — 0015 3.6 180° 0.875 1.470 — — — — 0021 3.6 180° 0.875 1.470 — — — — Swalwell et al. [27, 28] 0021 2.7 90° 0.93 1.55 0.105 0.151 Yes Yes
${\displaystyle {\left.Re\right._{c}}}$ Reynolds number based on chord length and free-stream velocity, ${\displaystyle {{\overline {C_{l}}},{\overline {C_{d}}}}}$ time-averaged lift and drag coefficients at α = 60° ${\displaystyle {\sigma \left[C_{l}\right],\sigma \left[C_{d}\right]}}$ standard deviation of lift and drag coefficients Spec force spectra available? ${\displaystyle {{\overline {C_{p}}}(x)}}$ surface pressure distribution available?
The choice of primary specific test case for this UFR is therefore the flow around a NACA0021 profile at Re = 2.7×105, α = 60° and with a spanwise domain size of Lz = 4c. The justification for this choice is given by the above reviews of previous computational and experimental studies:
• The chosen test case is the only one with unsteady experimental benchmark data [27, 28].
• A significant degree of scatter in the mean force coefficients is apparent between facilities, e.g. 12% for ${\displaystyle {\overline {C_{l}}}}$ at α = 60° (Table 1) and information about the relative fidelity of the experiments is absent. Therefore the experiment with the richest data is chosen (see previous point).
• Although it would be desirable to choose an experiment that has been conducted primarily for the validation of CFD, this is not the case for any of the available experiments.
• A large body of computational data is available for the chosen case, which in contrast to the experiments exhibits a low degree of scatter. This data is taken from recognised international comparison exercises with a rich variety of turbulence models assessed, good quality control, and systematic investigation of the principal test case uncertainties.
• In addition to the primary NACA0021 test case, earlier results from the NACA0012 test case will be drawn upon to increase the richness of the best practice advice (BPA). This is justifiable because of experimental evidence that the effects of Reynolds number and profile thickness are negligible in the range 25° < α < 180° [20].
Contributed by: Charles Mockett; Michael Strelets — CFD Software GmbH and Technische Universitaet Berlin; New Technologies and Services LLC (NTS) and St.-Petersburg State Polytechnic University
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# Explore Waterwave, Bigwave, and more!
### Explore related topics
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Pinterest • The world’s catalog of ideas
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# Tag Info
30
What you seem to be looking for is deniable authentication. This is actually a somewhat stronger property than what you're asking for: it guarantees that the recipient (let's call him Bob) cannot cryptographically convince anyone else that the sender (let's call her Alice) signed the message, even if he discloses all his private keys, simply because the ...
12
Lets say Alice wants to send Bob a sensitive message, she wants to prove to Bob that it came from her, but she doesn't want Bob to be able to prove that to anyone else. A MAC is a good way of doing this. If Alice and Bob share a MAC key (and only they have it) then Bob will know any message authenticated with that MAC key came from Alice, since he knows he ...
4
This was discussed by Coron in 1. You are actually asking why the random oracle can't just be some uncontrollable ideal random oracle. In fact Bellare and Rogaway when introduced their Full Domain Hash scheme (FDH) in the seminal works 2,3 used this uncontrollable random oracle to analyze the security reduction for FDH. The thing about using reductions ...
3
Don't use SHA-1. There's unlikely to be a substantive difference between the other choices, as far as you're concerned, except performance: SHA-256 is might be cheaper on 32-bit CPUs; SHA-384 and SHA-512 are cheaper on 64-bit CPUs. NIST P-256 is likely to be cheaper than NIST P-384 which is likely to be cheaper than NIST P-521. All of these choices ...
2
This diagram is not accurate. Hashing is not a separate step outside signing for the convenience of handling long messages; hashing is an integral part of signing that is necessary for security. And the verification algorithm does not always return a hash that you can compute separately: it only returns a boolean that indicates a valid signature or not. ...
2
In general case, $k^{-1}$ is equal to $x$ such that $x \cdot k=1$. In your question, to computing $11^{-1}$, you must find $x$ such that $x\cdot11=1 \pmod {8368}$. You can compute $x$ by using the extended Euclidean algorithm.
2
Suppose a signature is not an integer $s$ such that $s^e \equiv H(m) \pmod N$, but rather a pair of integers $(s, k)$ with $s < N$ and $k < N^{e - 1}$ such that $$s^e = H(m) + kN.$$ Then the verifier can verify this equation modulo a secret uniform random $v$-bit prime $r$, $$s^e \equiv H(m) + kN \pmod r.$$ There are $\pi(2^v) - \pi(2^{v-1})$ such ...
2
(This is to complement Avilan's answer on a more philosophical level.) In the random-oracle model (ROM for short) [BR], all parties are assumed to have oracle-access to a public random function $H$. The security of a protocol is then argued relative to this random oracle $H$, and then in practice $H$ is instantiated by an appropriate hash function (say, ...
1
Suppose you have a forgery procedure which takes a public key, calls SHA-256, interacts with an automatic PGP mail system, does some horrible computation, and returns an attempted forgery: import hashlib import smtplib def forge(pubkey): ... hashlib.sha256(m0) ... smtplib.sendmail(m1) ... return (forged_msg, forged_sig) We can take the text of ...
1
The words "controls"(in the question) and "manipulates"(in the paper) can be somehow misleading as to what is happening. Often in literature this is rather formulated as: emulates a random oracle, etc... One could quote the paper(with modifications) as follows: Given a forger $\mathcal{F}$ for the $GDH$ group $G$, we build an algorithm $\mathcal{A}$ that ...
1
If $l<k$, then the attack will be detected with a probability 1. If $l\ge k$, the probability of the attack not being detected is $$\frac{l}{n}\cdot \frac{l-1}{n-1}\cdots \frac{l-k+1}{n-k+1}\ge (\frac{l-k+1}{n-k+1})^k$$ Whether this is large enough or not depends on $n,k,l$. If the attacker does not know $k$, the best he can do is to choose $l=n-1$, ...
1
This problem is well defined for supply chain management, and generally you use a challenge-response. In RFID, you an use asymmetric or symmetric schemes to verify that a product is authentic. I give RFID as an example because you should be able find a lot of documentation on them. In the asymmetric case: A authentic product public key is on the ...
Only top voted, non community-wiki answers of a minimum length are eligible
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# GRE® Number Properties Practice
Concept: Number of factors of an integer.
This GRE quant practice question is a number properties quantitative comparison question. Concept tested: Computing number of factors of integers; using that concept finding the smallest positive integer with 7 factors.
#### Question: N is the smallest positive integer that has 7 factors.
Quantity A Quantity B
Number of factors of √N. Number of factors of (N-2).
1. Quantity A is greater
2. Quantity B is greater
3. The two quantities are equal
4. The relationship cannot be determined from the information given
Video Explanation will be added soon
### 3 steps to the answer
##### Use these hints to get the answer
1. Find the smallest positive integer that will have 7 factors.
2. Find the number of factors of $$sqrt{N}$. 3. Find the number of factors of$N - 2).
### Evaluate Quantity A
#### Step 1: Find the smallest positive integer that has 7 factors
The number of factors, 7, is a prime number. ∴ the only way to express 7 as a product of 2 numbers is 1 * 7.
If a number N can be prime factorized as ap * bq, where 'a' and 'b' are prime factors of N, number of factors of N = (p + 1)(q + 1)
So, any number that has 7 factors will have p = 0 and q = 6. i.e., the number will have only one prime factor.
2 is the smallest prime number. So, the smallest number that will have 7 factors is 26 = 64.
#### Step 2: Compute number of factors of $$sqrt{N}$ $\sqrt{64}$ = 8. Method: If a number N can be prime factorized as ap * bq, where 'a' and 'b' are prime factors of N, number of factors of N =$p + 1)(q + 1)
Prime factorize 8: 8 = 23.
Number of factors of 8 = (3 + 1) = 4.
Value of Quantity A is 4.
### Evaluate Quantity B
#### Compute number of factors of (N - 2)
N = 64. Therefore, (N - 2) = 62
Method: If a number N can be prime factorized as ap * bq, where 'a' and 'b' are prime factors of N, number of factors of N = (p + 1)(q + 1)
Prime factorize 62: 62 = 2 * 31
Number of factors of 62 = (1 + 1) * (1 + 1) = 4
Value of Quantity B is 4.
### The Comparison
Quantity A: Number of factors of $$sqrt{N}$ = 4 Quantity B: Number of factors of$N - 2) = 4
Both quantities are equal. Choice C is the answer.
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Journal article Open Access
# Implicit and Explicit Concept Relations in Deep Neural Networks for Multi-Label Video/Image Annotation
Markatopoulou, Foteini; Mezaris, Vasileios; Patras, Ioannis
### Citation Style Language JSON Export
{
"DOI": "10.1109/TCSVT.2018.2848458",
"author": [
{
"family": "Markatopoulou, Foteini"
},
{
"family": "Mezaris, Vasileios"
},
{
"family": "Patras, Ioannis"
}
],
"issued": {
"date-parts": [
[
2018,
6,
18
]
]
},
"abstract": "<p>In this work we propose a DCNN (Deep Convolutional Neural Network) architecture that addresses the problem of video/image concept annotation by exploiting concept relations at two different levels. At the first level, we build on ideas from multi-task learning, and propose an approach to learn conceptspecific representations that are sparse, linear combinations of representations of latent concepts. By enforcing the sharing of the latent concept representations, we exploit the implicit relations between the target concepts. At a second level, we build on ideas from structured output learning, and propose the introduction, at training time, of a new cost term that explicitly models the correlations between the concepts. By doing so, we explicitly model the structure in the output space (i.e., the concept labels). Both of the above are implemented using standard convolutional layers and are incorporated in a single DCNN architecture that can then be trained end-to-end with standard back-propagation. Experiments on four large-scale video and image datasets show that the proposed DCNN improves concept annotation accuracy and outperforms the related state of-the-art methods.</p>",
"title": "Implicit and Explicit Concept Relations in Deep Neural Networks for Multi-Label Video/Image Annotation",
"type": "article-journal",
"id": "1308778"
}
48
36
views
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Ask your own question, for FREE!
Mathematics
OpenStudy (anonymous):
Why is ds/dt of s=t/9t+2 = 2/(9t+2)^2 and not the derivative 9x^2+2?
OpenStudy (anonymous):
Basically, I answered the question with 9x^2+2, but I was wrong and I'm trying to figure out why. :(
OpenStudy (shadowfiend):
Well, when you have an equation divided by another equation, you have to apply the quotient rule or the chain rule. I prefer the chain rule, personally. So: $\frac{t}{9t + 2} = t^{-1}(9t + 2)$ If you apply the chain rule to that, the answer might make a bit more sense.
OpenStudy (shadowfiend):
Well, chain rule + product rule.
OpenStudy (anonymous):
So would t/5t+1 be equal to 2/(5t+1)^2?
OpenStudy (shadowfiend):
Heh, I totally messed the thing up there up. $\frac{t}{9t + 2} = t(9t + 2)^{-1}$
OpenStudy (shadowfiend):
No, it wouldn't derive to that. It would derive to 1/(5t + 1)^2.
OpenStudy (shadowfiend):
You do this by saying: $\frac{ds}{dt}t(5t+1)^{-1}$ \begin{align} (5t + 1)^{-1} + t(-1)(5t + 1)^{-2}(5)\\ \frac{1}{5t + 1} - \frac{5t}{(5t + 1)^2}\\ \frac{5t + 1}{(5t+1)^2} - \frac{5t}{(5t + 1)^2}\\ \frac{5t + 1 - 5t}{(5t + 1)^2}\\ \frac{1}{(5t + 1)^2} \end{align}
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# Power Analysis with Existing Data Set
I am new to this forum but have found several threads to be highly useful so am posing a question myself.
My data was collected (fish length = factor, fish mercury = response) from several rivers over several years for the purpose of environmental (mercury) monitoring.
What I would like to do is, using the data I have, perform power analysis to determine how many samples should be collected in the future to get the same results. The purpose is to recommend the least number of samples necessary (thus killing the least number of fish). Is this possible and does anyone have a recommendation on how to go about this?
• What sort of statistical software do you or can you use? – Dimitriy V. Masterov Apr 1 '13 at 20:59
• When you say you want "to get the same results" in the future what do you really mean? Is the monitoring to determine the level of mercury in the river, and the fish length is just a nuisance factor to be controlled for? So does that mean what you are looking for is a large enough sample size to detect an increase in X% of mercury, shown in an average increase in fish mercury after controlling for fish length? This is certainly possible. – Peter Ellis Apr 1 '13 at 23:22
• Dimitriy: I can use SPSS or GraphPad Prism. I have access to almost any statistical software that is user-friendly at the university. I am, however, not able to use R well so would prefer to avoid it. – AshP Apr 1 '13 at 23:47
• Peter: Sorry, that was not phrased well. I want to collect a large enough sample size to detect an effect and perform statical tests (the specific test may vary in the future depending who does what with the data). We collect fish using a length-based approach since mercury is thought to be a function of length. Mercury may not increase each year, it is variable, but yes, I am looking for changes (trends) in fish mercury after controlling for length. – AshP Apr 1 '13 at 23:53
Step 1: Estimate the size of the effect you have gotten in your current data (e.g., r, Cohen's D)
Step 2: Get G*Power
Step 3: Using G*Power, calculate the required sample size given the size of the effect you have, the level of alpha (.05 usually), and the amount of power you want (.80 is common).
If you outline the specific type of analysis you did and want to do, I can guide you a bit further.
• Thank you! Are these tests only possible to perform in R? What do you mean by "size of the effect"? And what do you mean by "specific type of analysis"? Do my previous comments answer that question? – AshP Apr 1 '13 at 23:55
• This would be an approximation, but seems to ignore the fact that that effect size is an estimate; the actual effect size in the population could be smaller -- if you want to have a particular level of confidence in detecting the actual effect size in the population, the uncertainty in the effect size estimate would need to be considered. It would lead to a slightly larger sample size for a given power. – Glen_b Apr 2 '13 at 4:04
• Lets say for example you are interested in looking at how length correlates with mercury levels. You already have data on this, so you can determine the approximate size of the effect between these variables. Lets say the correlation is .50. GPower tell me that you need approximately 21 cases (fish) to have an 80% chance to statistically support this relationship in a future sample. This is a pretty layman explanation and perhaps a bad example, but that is the crux. You can calculate for t-tests, ANOVAs, etc. in GPower. – Behacad Apr 2 '13 at 13:55
• Also, I will mention that G*Power can help you identify the size of the effect using values often outputted by SPSS and other programs. If you say what specific statistical analysis you are doing (e.g., t-test, ANOVA, regression, correlation, chi-square) we can guide you more specifically. – Behacad Apr 2 '13 at 20:25
If you have access to Stata, you can trying using the user-written command powerreg. It does multiple linear regression power analysis. There's a nice example of how to use it at the UCLA ATS site.
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# Use the known area of a circle to find the value of the integral
## Main Question or Discussion Point
Someone plz show me how to do the problem below, thanks very much.
1)***Use the known area of a circle to find the value of the integral
integral from -a to a of the function sqrt(a^2-x^2)dx.
2)***Then use the result of this integral to find the enclosed area of (x^2)/(a^2)+(y^2)/(b^2)= 1, a>b>0.
Plz show me how to integrate #1
Last edited:
Use the substitution x=asinu so that sqrt(a2-x2)=acosu.
Originally posted by gigi9
***Then use the result of this integral to find the enclosed area of (x^2)/(a^2)+(y^2)/(b^2), a>b>0.
The enclosed area of what?
Hurkyl
Staff Emeritus
Gold Member
***Use the known area of a circle to find the value of the integral
integral from -a to a of the function sqrt(a^2-x^2)dx. ***
You know that the definite integral of a positive function is the area between its graph and the x-axis right? What is the graph of sqrt(a^2-x^2)?
still confused..explain more plz
HallsofIvy
Homework Helper
You seem to have serious problems with basic concepts- as illustrated by your saying "Then use the result of this integral to find the enclosed area of (x^2)/(a^2)+(y^2)/(b^2), a>b>0."
I presume that you copied this from some problem but you even copied wrong. "(x^2)/(a^2)+(y^2)/(b^2)" does not enclose anything- it is not a graph nor a function nor an equation. I suspect that you book had "(x^2)/(a^2)+(y^2)/(b^2)= 1", the equation of an ellipse.
As for the first problem: If you are expected to be able to do integrals, then you should already know that a basic interpretation of "integral" is "area under a curve". The function y= sqrt(a^2-x^2)dx is the upper half of the circle x^2+ y^2= a^2 (you can see that by squaring both sides of the given equation). Since the circle has area πa2, the semi-circle has area πa2/2 and that is the value of the integral of the function.
Now that you know that integral, solve (x^2)/(a^2)+(y^2)/(b^2)= 1 for y and apply that knowledge.
plz show me how to integrate the 1st one...and how to find the enclosed area of the second one plz...(maybe the first few step or something to get me started...) Thanks a lot.
HallsofIvy
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When we record data, we often obtain data that can be approximated nicely by a straight line. If this is the case, it tremendously simplifies our description of the dataset: we do not need to know every simgle datapoint any more, but only the two coefficients defining the best-fit line to the data points.
To obtain the best-fit line to noisy data, we want to find the model coefficients “k” and “d” that define our data-model “y=k*x + d”.
The following chapter describes how to obtain the model coefficients for simple linear models, how accurately we know these coefficients, and how this concept can be used to describe much more complex patterns, such as polynomials, circles, etc.
# Fully determined linear model¶
A linear model describes a linear relationship between a dependent variable and a number of independent variables. In the simplest case, there is one dependent variable $$y$$ and one independent variable $$x$$. We can write the linear model as
$\label{eq:line} y = mx+c$
where $$m$$ and $$c$$ are the two free parameters. This is a line in a plane, with $$m$$ the slope, and $$c$$ the y-intercept. If we have two different data points relating the dependent and independent variables, then these two free parameters are fully determined. For example, if $$P_1=(x_1 / y_1)$$ and $$P_2=(x_2 / y_2)$$, then we can calculate that
$m = \frac{{\Delta y}}{{\Delta x}} = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\,and\,c = \frac{{{y_1}{x_2} - {x_1}{y_2}}}{{{x_2} - {x_1}}}$
In the general case, there is one dependent variable $$y$$ and $$k$$ independent variables $$x_i$$. The linear model then becomes
$y = c + \sum\limits_{d = 1}^k {{m_d}{x_d}}$
where $$c, m_1, ..., m_k$$ are the $$k+1$$ free parameters. If we have $$k+1$$ different data points that lie exactly on an k-dimensional hyperplane, then the free parameters are fully determined.
## Reminder: Lines in a plane¶
A line can either be represented by its slope and y-intercept, as shown above. Alternatively, it can be characterised by a point on the line, $$\vec{x_1}$$ , and the line’s normal vector, $$\vec{n}$$ :
$\vec n \cdot \vec x = \vec n \cdot {\vec x_1}$
or
$\begin{split}ax + by = c,\,with\,\vec n = \left( {\begin{array}{{20}{c}} a\\ b \end{array}} \right)\end{split}$
# Overdetermined linear model¶
Mostly, we have more data points than free parameters. In this case, the model is overdetermined by the data, so that it is not possible to fit the model exactly to all of the data points. Instead, we try to find a best fit model, which is the model with the minimum fitting error. Often, the fitting error is taken to be the least squares estimator.
## Residual¶
The difference between a data value and the corresponding model value is referred to as a residual. If we have a data point where the dependent variable has the value $$y_i$$ and the independent variables have value $$x_{id}$$, then the residual $$e_i$$ for the general linear model is
${e_i} = {y_i} - \left( {c + \sum\limits_{d = 1}^k {{m_d}{x_{id}}} } \right)$
## Least squares estimators¶
If there are $$n$$ data points, the sum of the squared residuals is
$E({x_{id}},{y_i},c,{m_d}) = \sum\limits_{i = 1}^n {e_i^2} = \sum\limits_{i = 1}^n {{{\left( {{y_i} - \left( {c + \sum\limits_{d = 1}^k {{m_d}{x_{id}}} } \right)} \right)}^2}}$
The least squares estimators are the values of $$c$$ and $$m_1, ..., m_k$$ that minimise E. To determine the value of the least squares estimators $$\hat c$$ and $${\hat m_1},...,{\hat m_k}$$ , it is necessary to locate the minimum of E by finding where the following partial derivatives are zero:
$\begin{split}\begin{array}{l} 0 = {\left. {\frac{{\partial E}}{{\partial c}}} \right|_{\hat c,{{\hat m}_d}}} = - 2\sum\limits_{i = 1}^n {\left( {{y_i} - \left( {\hat c + \sum\limits_{d = 1}^k {{{\hat m}_d}{x_{id}}} } \right)} \right)} \\ 0 = {\left. {\frac{{\partial E}}{{\partial {m_p}}}} \right|_{\hat c,{{\hat m}_d}}} = - 2\sum\limits_{i = 1}^n {\left( {{y_i} - \left( {\hat c + \sum\limits_{d = 1}^k {{{\hat m}_d}{x_{id}}} } \right)} \right)} {x_{ip}}\,\,\,for\,all\,p \end{array}\end{split}$
This linear system of equations can then be solved to find the values of the least squares estimators.
## Reminder: Ordinary Least Squares¶
The method of ordinary least squares can be used to find an approximate solution to overdetermined systems. For the system $$\mathbf{A} \cdot \vec{p} = \vec{y}$$, the least squares formula is obtained from the problem
$\min_{p}\Vert \mathbf{A} \cdot \vec{p} - \vec{y} \Vert,$
the solution of which can be written with the normal equations,
$\vec{p} =(\mathbf{A}^{\mathrm{T}}\mathbf{A})^{-1}\mathbf{A}^{\mathrm{T}} \cdot \vec{y},$
where $$\mathrm{T}$$ indicates a matrix transpose, provided $$(\mathbf{A}^{\mathrm{T}}\mathbf{A})^{-1}$$ exists (that is, provided $$A$$ has full column rank). With this formula an approximate solution is found when no exact solution exists, and it gives an exact solution when one does exist.
# Finding least squares estimators with MATLAB¶
## Linear model with no intercept term¶
Let us start with the case where there is no intercept term (i.e., $$c$$ is zero). Our model is
$y \approx \sum\limits_{d = 1}^k {{m_d}{x_d}}$
X is an $$n * k$$ matrix made of the column vectors $$x_1, ... , x_k$$, $$m$$ is a $$k * 1$$ column vector containing the free parameters $$m_1, ... , m_k$$, and $$y$$ is an $$n * 1$$ column vector, then we can rewrite the model as
$X \cdot m \approx y$
The least squares estimator for $$m$$ can be obtained easily in Matlab by reshaping the equation:
m_estimator = X\y;
Note that the “\” is a backslash and not a normal divide. Alternatively, one can use the command regress:
m_estimator = regress(y, X);
## Linear model with intercept¶
Now let us address the general linear model with intercept term:
$y \approx c * \sum\limits_{d = 1}^k {{m_d}{x_d}}$
We rewrite this in matrix form in the following way:
$\label{eq:regressMatrix} \vec y \approx {\bf{M}} \cdot \vec p$
where
$\begin{split}\label{eq:regressionFit} {\bf{M}} = \left[ {\begin{array}{{20}{c}} {{x_{11}}}&{{x_{12}}}& \cdots &{{x_{1k}}}&1\\ {{x_{21}}}&{{x_{22}}}& \cdots &{{x_{2k}}}&1\\ \vdots & \vdots &{}& \vdots & \vdots \\ {{x_{n1}}}&{{x_{n2}}}& \cdots &{{x_{nk}}}&1 \end{array}} \right],\,\,p = \left[ {\begin{array}{{20}{c}} {{m_1}}\\ {{m_2}}\\ \vdots \\ {{m_k}}\\ c \end{array}} \right],\,\,y = \left[ {\begin{array}{{20}{c}} {{y_1}}\\ {{y_2}}\\ \vdots \\ {{y_n}} \end{array}} \right]\end{split}$
In Matlab, we can solve this in the same way as before, namely The matrix M is sometimes called Design Matrix. In Matlab, we can solve this in the same way as before:
M = [X ones(size(y))];
p_estimator = M\y;
Again, it is possible to use the function regress:
p_estimator = regress(y, M);
## Example 1: Fitting a line¶
For a line-fit, the equation for general linear models is reduced to
$\begin{split}\left[ {\begin{array}{{20}{c}} {{y_1}}\\ {{y_2}}\\ {{y_3}}\\ {...} \end{array}} \right] = \left[ {\begin{array}{{20}{c}} {{x_1}}&1\\ {{x_2}}&1\\ {{x_3}}&1\\ {...}&{...} \end{array}} \right] \cdot \left[ {\begin{array}{{20}{c}} k\\ d \end{array}} \right]\end{split}$
% Generate a noisy line, with an slope of 0.5
% and a y-intercept of -30
x = 1:100;
y = -30 + 0.5*(x + 5*randn(size(x)));
% Plot the line
plot(x,y);
line(xlim, [0 0], 'LineStyle', '--');
% Calculate the best-fit line
M = [x' ones(length(x),1)];
p_estimator = M\y'; % or "p_estimator = regress(y', M);"
slope = p_estimator(1) % "m"
yintercept = p_estimator(2) % "c"
An alternative way to fit a line would be to see the line as a $$1^{st}$$ order polynomial, and use the Matlab function polyfit:
p = polyfit(x,y,1);
y_fit = polyval(p, x);
## Example 2: Fitting a polynomial¶
We can use the same approach to fit a polynomial curve to the data. For example, for a quadratic relationship between x and y we have
$y = a*x^2 + b*x + c$
Written in matrix form, this gives
$\begin{split}\left[ {\begin{array}{{20}{c}} y_1\\ y_2\\ y_3\\ {...} \end{array}} \right] = \left[ {\begin{array}{{20}{c}} x_1^2 & x_1 & 1\\ x_2^2 & x_2 & 1\\ x_3^2 & x_3 & 1\\ {...} & {...} & {...} \end{array}} \right] \cdot \left[ {\begin{array}{{20}{c}} a\\ b\\ c \end{array}} \right]\end{split}$
With this we have brought the problem into the form $$\vec{y} \approx \bf{M} \cdot \vec{p}$$, and we can solve it in the same way as above.
## Example 3: Fitting a sine-wave¶
If we have a sinusoidal oscillation, where we know the frequency $$\omega$$, and where we want to know the amplitude, the phase delay, and the offset, our function can be written as
$x = offset \cdot 1 + amplitude \cdot \sin (\omega t + \delta )$
Note that here offset and amplitude appear as linear parameters - but the phase, $$\delta$$, does not. But this can be solved by expressing $$sin(\omega*t+\delta)$$ by the sum of a sine- and a cosine wave:
$\label{eq:sinFit} x = offset \cdot 1 + a \cdot \sin (\omega t) + b \cdot \cos (\omega t)$
From these parameters, we can find the amplitude and the phase:
$\begin{split}\begin{array}{l} amplitude = \sqrt {{a^2} + {b^2}} \\ \delta = {\tan ^{ - 1}}\left( {\frac{b}{a}} \right) \end{array}\end{split}$
Now all the parameters that we want to know appear in a linear (!!) relationship, and we can put together our matrix M as
$\begin{split}{\bf{M}} = \left[ {\begin{array}{{20}{c}} 1&{\sin (\omega \cdot {t_1})}&{\cos (\omega \cdot {t_1})}\\ 1&{\sin (\omega \cdot {t_2})}&{\cos (\omega \cdot {t_2})}\\ 1&{\sin (\omega \cdot {t_3})}&{\cos (\omega \cdot {t_3})}\\ {...}&{...}&{...} \end{array}} \right]\end{split}$
and $$\vec{p}$$, which is here given by $$\vec p = \left( {\begin{array}{*{20}{c}} {offset}\\ a\\ b \end{array}} \right)$$ can again be obtained from the solution of the linear model.
% Time
t = 0:0.1:8*pi;
% Set the parameters
freq = 0.5;
offset = 3;
delta = 45*pi/180;
amplitude = 2;
omega = 2*pi*freq;
% Simulate and plot noisy sine-wave:
y = offset + amplitude * sin(omega*t + delta) + randn(size(t));
plot(t,y)
% Fit the data
M = [ones(length(t),1) sin(omega*t') cos(omega*t')];
p = M\y';
% Extract the coefficients from the fit
found.offset = p(1);
found.amp = sqrt(p(2)^2 + p(3)^2);
found.delta = atan2(p(3), p(2))*180/pi;
% Superpose the original data with the fit
hold('on');
found.y = found.offset + found.amp*sin(omega*t + found.delta*pi/180)
plot(t, found.y, 'r')
legend('original', 'fit');
xlabel('Time [s]');
ylabel('Data');
shg
Sine-Fit
## Example 3: Fitting a circle¶
The same concept can surprisingly even be extended to find the best-fit circles to data. To do so, we rearrange the general equation for a circle:
$\begin{split}\begin{array}{c} {(x - {x_0})^2} + {(y - {y_0})^2} = {r^2}\\ {x^2} - 2x{x_0} + x_0^2 + {y^2} - 2y{y_0} + y_0^2 = {r^2}\\ 2x \cdot {x_0} + 2y \cdot {y_0} + 1 \cdot ({r^2} - x_0^2 - y_0^2) = {x^2} + {y^2} \end{array}\end{split}$
This gives us $${x^2} + {y^2} = {\bf{M}} \cdot \vec p$$ , with
$\begin{split}{\bf{M}} = \left[ {\begin{array}{{20}{c}} {2{x_1}}&{2{y_1}}&1\\ {2{x_2}}&{2{y_2}}&1\\ {...}&{...}&{...} \end{array}} \right]\end{split}$
Again, this is just a linear fit! From this we get
$\begin{split}\begin{array}{l} {x_0} = p(1)\\ {y_0} = p(2)\\ r = \sqrt {p(3) + x_0^2 + y_0^2} \end{array}\end{split}$
## Confidence Interval for Linear Regressions¶
Let us now generalize this concept of confidence intervals to our linear regression, and add some more information about the basic assumptions underlying the calculation of confidence intervals. In the previous section, we saw that it is possible to describe a linear trend via the least squares estimators. However, measured data always have errors superimposed on the basic trend in which we are interested. This error leads to uncertainty in the estimators.
To deduce something about the uncertainty in the estimators, we have to make assumptions about the noise in the data. Some common assumptions that are made are:
• we know the independent variables $$x_1, ... , x_k$$ exactly
• the residuals are roughly normally distributed
• the noise is only in the dependent variable y
• the residuals are independent of the values of $$x_1, ... , x_k$$
If any of these assumptions does not hold, then the confidence interval must be calculated differently.
The Matlab function regress can give you both the least squares estimators and confidence intervals for the parameters:
[B, BINT] = regress(y, X);
where B indicates the least squares estimators $$\hat m$$, and BINT the 95% confidence interval for $$\hat m$$. Remember, though, that Matlab calvculates confidence intervals as described above, so check the assumptions first.
To specify a different confidence level, use the form
[B, BINT] = regress(y, X, alpha);
Note that the confidence interval widens as the required confidence level increases.
## Relationship to hypothesis testing¶
A confidence interval for a parameter can be used directly to test the corresponding null hypothesis. With the confidence intervals described above, the null hypothesis is:
The parameter is not significantly different from zero.
If the confidence interval contains zero, then it is not possible to reject the null hypothesis at the used confidence level; in other words, there is then no statistical evidence to suggest that the parameter is significantly different from zero.
## Line of best fit with no intercept term¶
If we use regress to fit a linear model with no intercept term to data in Fig. [fig:lineThroughZero], we obtain $$B = 0.755$$ and $$BINT = [0.748 0.761]$$.
Sample noisy data with a line of best fit passing through the origin.
This means that the slope of the best fit line (i.e., $$k$$) is most likely to be $$0.755$$, and lies with 95% probability between $$0.748$$ and $$0.761$$.
## Line of best fit with intercept term¶
If we use regress to fit a linear model with intercept term to the data in Fig. [fig:lineWithOffset],
B is $$\left[ {\begin{array}{*{20}{c}} {0.751}\\ {20.228} \end{array}} \right]$$ and BINT is $$\left[ {\begin{array}{*{20}{c}} {0.739}&{0.764}\\ {19.483}&{20.974} \end{array}} \right]$$ .
Sample noisy data, where the line of best fit does not pass through the origin.
This means that the slope of the best fit line (corresponding to $$m$$) is most likely $$0.751$$, and lies with 95% probability between $$0.739$$ and $$0.764$$; and that the offset of the line (corresponding to $$c$$) is $$20.228$$, and lies with 95% probability between $$19.483$$ and $$20.974$$.
## Significance¶
If both confidence for the slope are larger than 0, we speak of a significant increase in the data. If also both 99.9% confidence intervals are above 0, we call this a highly significant increase. Similarly, if both confidence intervals are below zero, our data show a significant or a highly significant decrease.
On the other hand, if the confidence intervals overlap zero, we cannot claim that our data are in- or decreasing, even if the slope is positive or negative, respectively.
# Correlation Coefficient and Coefficient of Determination¶
## Correlation Coefficient¶
The correlation coefficient r is a measure of the linear correlation (or dependence) between two variables $$x$$ and $$y$$, giving a value between +1 and -1 inclusive, where 1 is total positive correlation, 0 is no correlation, and -1 is total negative correlation. For sample data $$x_i$$ and $$y_i$$, r indicates the change in $$x$$ multiplied by the change in $$y$$, normalized by the relative spread of $$x$$ and $$y$$:
$\label{eq:corrcoef} r = \frac{\sum ^n _{i=1}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum ^n _{i=1}(x_i - \bar{x})^2} \sqrt{\sum ^n _{i=1}(y_i - \bar{y})^2}}$
Note that the correlation coefficient is symmetric in $$x$$ and $$y$$.
For linear regression, the square of the correlation coefficient r equals the coefficient of determination.
## Coefficient of Determination¶
The coefficient of determination tells us how well the fitted data account for the raw data. It makes use of the fact that
$SST = SSR + SSE$
where SST* is the sum of the squares of the total deviation (Figure below, left), SSR the sum of the squares of the regression, and SSE the sum of the squares of the errors (Figure below, right):
$\begin{split}SST &= \sum\limits_i (y_i - \bar{y})^2 \\ SSR &= \sum\limits_i (\hat{y}_i - \bar{y})^2 \\ SSE &= \sum\limits_i (y_i - \hat{y}_i)^2\end{split}$
The better the linear regression (on the right) fits the data in comparison to the simple average (on the left graph), the closer the value of $$r^2$$ is to one. The areas of the blue squares represent the squared residuals with respect to the linear regression. The areas of the red squares represent the squared residuals with respect to the average value (from Wikipedia).
$$\bar{y}$$ is the average value, and $$\hat{y}$$ is the value of the fitted data that corresponds to a given $$x_i$$ value.
The coefficient of determination is defined as the amount of the sum of squares that can be explained by the regression:
$r^2 = \frac{SSR}{SST} = 1 - \frac{SSE}{SST}$
As an example, if $$r^2 = 0.7$$, this would tell us that approximately seventy percent of the variation in the response can be explained by the line fit.
To illustrate a range of different fits, three data sets with different amounts of noise are plotted in the Figure below. All three data sets have roughly the same line of best fit, but the coefficients of determination differ.
Data with different levels of noise.
In Matlab, the correlation coefficient r can be found by using the function corr.
x = 1:100;
y = 10 + 0.5*x + randn(size(x));
plot(x,y,'.')
corr(x', y')
produces a value of approximately r = 0.997.
# The Next Level¶
## Fitting of nonlinear functions¶
So far, we have only considered a linear relationship between the data and the parameters. Note that even tasks such as fitting a sine-function with an offset and a phase can be expressed with linear relationships (see Example 2: Fitting a sine-wave). However, often, the relationship will be visibly nonlinear, and cannot be fit with a linear model. In this case, it is necessary to select a nonlinear model for the data, and then attempt to fit the parameters of the model.
For example, for the data in the figure below, the falling part of the curve might reflect some physical process that is naturally modelled with an exponential decay.
Plot of a variable that decays over time.
Since an exponential decay is fully defined by the starting value and time, the value of the asymptotic minimum, and the half-life, one way of fitting this curve would be to estimate these parameters independently.
Alternatively, one could use more sophisticated methods that attempt to estimate all of the model parameters simultaneously. Such methods can be found in Matlab in the Curve Fitting Toolbox or in the Optimization Toolbox. Often, though, the methods run more efficiently when the user provides a good estimate of the parameters as a starting point.
An important tip is to choose a model/function that is likely to fit the data well, and if you have a choice, to reduce the number of parameters that must be estimated.
### Example:¶
For fitting an exponential decay, you first have to determine where the decay starts. If the data are not too noisy, this can be automated by finding e.g. the first value clearly below the maximum. Clearly can be e.g. 10% below the maximum
tStart = min(find(x<max(x)*threshold));
While the quality of the fit depends on many factors, e.g. the quality of the data, the appropriateness of the model, and the starting values, the Matlab Curve Fitting Toolbox can sometimes make things remarkably easy. For example, to fit an exponential decay to an offset can be found with
% Generate dummy data
fitTime = (0:0.01:20)';
fitVal = 3 + 4*exp(-fitTime/5)+randn(size(fitTime));
% Make the fit
f = fittype('offset + amp*exp(-x/tau)');
options = fitoptions(f);
options.StartPoint = [1,1,1];
fitted = fit(fitTime,fitVal,f, options)
fitted =
General model:
fitted(x) = offset + amp*exp(-x/tau)
Coefficients (with 95% confidence bounds):
amp = 3.964 (3.79, 4.137)
offset = 3.01 (2.899, 3.122)
tau = 5.148 (4.565, 5.732)
# Exercises¶
## Exercise 1: Line Fits¶
• Read in the data from https://github.com/thomas-haslwanter/BioSignalAnalysis/blob/master/Data/co2.txt.
• Fit a line to the last 25 years of carbon emissions from “fossil fuels and cement”, using “polyfit”. Thereby, choose the values for the first datapoint to be (0/0).
• Fit a quadratic curve to the same data, using “polyfit”.
• Plot original data, line, and quadratic curve.
## Exercise 2: Confidence Intervals¶
Use the same data as in Exercise 1, but now also determine the 95% confidence intervals. Answer the following 2 questions:
• When you fit a line, is the slope of the line significantly rising?
• When you fit a quadratic curve, is the quadratic contribution significant?
Note: When the highest order term is determined, then all lower order terms are also included. If for instance we fit a fifth order polynomial, and only the cubic term is significant, then we would omit the higher order nonsignificant terms, but retain those terms of smaller order than the cubic.
## Exercise 3: Linear Circle Fit¶
All the data that you need for this and the next exercises are in https://github.com/thomas-haslwanter/BioSignalAnalysis/blob/master/Data/noisyStuff.mat. “noisyCircle” contains the x/y values of a noisy circle.
Write a function that takes these data and determines the center of the best-fit circle-center (x0/y0) and the radius r.
## Exercise 4: Nonlinear Exponential Decay¶
“noisyExp” contains the t/x values of a noisy function that exponentially decays to a constant offset.
Write a function that takes these data and calculates the best-fit Amplitude, Offset, and Decay-time for this decay.
## Exercise 5: Butterworth Filter¶
“noisySine” contains the t/x values of a noisy sine wave.
Smooth these data with a Savitzky-Golay filter, and superpose the filtered data with the original data. The MATLAB command butter provides the [b,a] coefficients of an IIR filter corresponding to Butterworth lowpass filter. Find the coefficients for a low-pass filter with a corner frequency of 0.5 Hz, and apply this IIR filter to the noisy sine wave. Compare the output with the output of the Savitzky-Golay filter.
## Exercise 6: Confidence Intervals¶
“noisyLine” contains the t/x values of a very noisy line.
A linear regression fit indicates that the data can be fit with a line (y = k*x+d), with k=0.1653.
Are the data significantly increasing?
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